Introduction to Probability 1st Blitzstein Solution Manual

Introduction to Probability 1st Blitzstein
Solution Manual download
https://testbankmall.com/product/introduction-to-probability-1st-
blitzstein-solution-manual/
Find test banks or solution manuals at testbankmall.com today!

We believe these products will be a great fit for you. Click
the link to download now, or visit testbankmall.com
to discover even more!
Introduction to Probability with Texas Hold em Examples
1st Schoenberg Solution Manual
https://testbankmall.com/product/introduction-to-probability-with-
texas-hold-em-examples-1st-schoenberg-solution-manual/
Probability Foundations for Engineers 1st Nachlas Solution
Manual
https://testbankmall.com/product/probability-foundations-for-
engineers-1st-nachlas-solution-manual/
Introduction to Intelligence Studies 1st Jensen III
Solution Manual
https://testbankmall.com/product/introduction-to-intelligence-
studies-1st-jensen-iii-solution-manual/
Test Bank for Introduction to General, Organic and
Biochemistry, 10th Edition, Frederick A. Bettelheim,
William H. Brown, Mary K. Campbell, Shawn O. Farrell, Omar
Torres
https://testbankmall.com/product/test-bank-for-introduction-to-
general-organic-and-biochemistry-10th-edition-frederick-a-bettelheim-
william-h-brown-mary-k-campbell-shawn-o-farrell-omar-torres/

Organizational Behavior Human Behavior at Work 14th
Edition Newstrom Test Bank
https://testbankmall.com/product/organizational-behavior-human-
behavior-at-work-14th-edition-newstrom-test-bank/
Test Bank for Death and Dying, Life and Living 8th Edition
Corr
https://testbankmall.com/product/test-bank-for-death-and-dying-life-
and-living-8th-edition-corr/
Cengage Advantage American Foreign Policy and Process 6th
Edition McCormick Test Bank
https://testbankmall.com/product/cengage-advantage-american-foreign-
policy-and-process-6th-edition-mccormick-test-bank/
Test Bank for Fundamentals of Statistics 5th Edition by
Sullivan
https://testbankmall.com/product/test-bank-for-fundamentals-of-
statistics-5th-edition-by-sullivan/
Business Law Today Comprehensive 11th Edition Miller Test
Bank
https://testbankmall.com/product/business-law-today-
comprehensive-11th-edition-miller-test-bank/

Test Bank for Concepts in Biology 14th Edition by Enger
https://testbankmall.com/product/test-bank-for-concepts-in-
biology-14th-edition-by-enger/

4. Fred is answering a multiple-choice problem on an exam, and has to choose one of n
options (exactly one of which is correct). Let K be the event that he knows the answer,
and R be the event that he gets the problem right (either through knowledge or through
luck). Suppose that if he knows the right answer he will definitely get the problem right,
but if he does not know then he will guess completely randomly. Let P (K) = p.
29

30
(a) Find P (K|R) (in terms of p and n).
(b) Show that P (K|R) ≥ p, and explain why this makes sense intuitively. When (if ever)
does P (K|R) equal p?
Solution:
(a) By Bayes’ rule and the law of total probability,
P(R|K)P(K) p
P (K|R) =
P (R|K)P (K) + P (R|Kc)P (Kc)
=
p + (1 − p)/n
.
(b) For the extreme case p = 0, we have P (K|R) = 0 = p. So assume p > 0. By the
result of (a), P (K|R) ≥ p is equivalent to p + (1 − p)/n ≤ 1, which is a true statement
since p + (1 − p)/n ≤ p + 1 − p = 1. This makes sense intuitively since getting the
question right should increase our confidence that Fred knows the answer. Equality
holds if and only if one of the extreme cases n = 1, p = 0, or p = 1 holds. If n = 1, it’s
not really a multiple-choice problem, and Fred getting the problem right is completely
uninformative; if p = 0 or p = 1, then whether Fred knows the answer is a foregone
conclusion, and no evidence will make us more (or less) sure that Fred knows the answer.
5. Three cards are dealt from a standard, well-shuffled deck. The first two cards are flipped
over, revealing the Ace of Spades as the first card and the 8 of Clubs as the second card.
Given this information, find the probability that the third card is an ace in two ways:
using the definition of conditional probability, and by symmetry.
Solution: Let A be the event that the first card is the Ace of Spades, B be the event
that the second card is the 8 of Clubs, and C be the event that the third card is an ace.
By definition of conditional probability,
P (C|A, B) =
P (C, A, B)
=
P (A, B)
P (A, B, C)
.
P (A, B)
By the naive definition of probability,
P (A, B) =
50!
=
1
and
52! 51 · 52
P (A, B, C) =
3 ·49!
=
3
.
So P (C|A, B) = 3/50.
52! 50 · 51 · 52
A simpler way is to see this is to use symmetry directly. Given the evidence, the third
card is equally likely to be any card other than the Ace of Spades or 8 of Clubs, so it
has probability 3/50 of being an ace.
6. A hat contains 100 coins, where 99 are fair but one is double-headed (always landing
Heads). A coin is chosen uniformly at random. The chosen coin is flipped 7 times, and it
lands Heads all 7 times. Given this information, what is the probability that the chosen
coin is double-headed? (Of course, another approach here would be to look at both sides
of the coin—but this is a metaphorical coin.)
Solution: Let A be the event that the chosen coin lands Heads all 7 times, and B be the
event that the chosen coin is double-headed. Then
P(A|B)P(B) 0.01 128
P (B|A) =
P (A|B)P (B) + P (A|Bc)P (Bc)
=
0.01 + (1/2)7 · 0.99
=
227
≈ 0.564.

227
|
Conditional probability 31
7. A hat contains 100 coins, where at least 99 are fair, but there may be one that is double-
headed (always landing Heads); if there is no such coin, then all 100 are fair. Let D be
the event that there is such a coin, and suppose that P (D) = 1/2. A coin is chosen
uniformly at random. The chosen coin is flipped 7 times, and it lands Heads all 7 times.
(a) Given this information, what is the probability that one of the coins is double-
headed?
(b) Given this information, what is the probability that the chosen coin is double-
headed?
Solution:
(a) Let A be the event that the chosen coin lands Heads all 7 times, and C be the event
that the chosen coin is double-headed. By Bayes’ rule and LOTP,
P (D|A) =
P(A|D)P(D)
.
P (A|D)P (D) + P (A|Dc)P (Dc)
We have P (D) = P (Dc
) = 1/2 and P (A|Dc
) = 1/27
, so the only remaining ingredient
that we need to find is P (A|D). We can do this using LOTP with extra conditioning
(it would be useful to know whether the chosen coin is double-headed, not just whether
somewhere there is a double-headed coin, so we condition on whether or not C occurs):
P (A|D) = P (A|D, C)P (C|D) + P (A|D, Cc
)P (Cc
|D) =
1
+
1
99
· .
Plugging in these results, we have
227
100 27 100
P (D|A) =
327
= 0.694.
(b) By LOTP with extra conditioning (it would be useful to know whether there is a
double-headed coin),
P (C|A) = P (C|A, D)P (D|A) + P (C|A, D
c
)P (D
c
|A),
with notation as in (a). But P (C|A, Dc
) = 0, and we already found P (D|A) in (a). Also,
P (C|A, D) = 128
, as shown in Exercise 6 (conditioning on D and A puts us exactly in
the setup of that exercise). Thus,
128 227 128
P (C|A) =
227
·
327
=
327
≈ 0.391.
8. The screens used for a certain type of cell phone are manufactured by 3 companies,
A, B, and C. The proportions of screens supplied by A, B, and C are 0.5, 0.3, and
0.2, respectively, and their screens are defective with probabilities 0.01, 0.02, and 0.03,
respectively. Given that the screen on such a phone is defective, what is the probability
that Company A manufactured it?
Solution: Let A, B, and C be the events that the screen was manufactured by Company
A, B, and C, respectively, and let D be the event that the screen is defective. By Bayes’
rule and LOTP,
P(D A)P(A)
P (A|D) =
P (D|A)P (A) + P (D|B)P (B) + P (D|C)P (C)
=
0.01 ·0.5
0.01 · 0.5 + 0.02 · 0.3 + 0.03 · 0.2
≈ 0.294.

32
P (B)
2
2.
j ) = 0.3 for j = 1, 2, since if Fred falls
(a) We need to find P (A3|A1) and P (A3|Ac
9. (a) Show that if events A1 and A2 have the same prior probability P (A1) = P (A2),
A1 implies B, and A2 implies B, then A1 and A2 have the same posterior probability
P (A1|B) = P (A2|B) if it is observed that B occurred.
(b) Explain why (a) makes sense intuitively, and give a concrete example.
Solution:
(a) Suppose that P (A1) = P (A2), A1 implies B, and A2 implies B. Then
P (A1|B) =
P (A1, B)
=
P (B)
P (A1)
=
P (B)
P (A2)
=
P (B)
P (A2, B)
= P (A |B).
(b) The result in (a) makes sense intuitively since, thinking in terms of Pebble World,
observing that B occurred entails restricting the sample space by removing the pebbles
in Bc
. But none of the removed pebbles are in A1 or in A2, so the updated probabilities
for A1 and A2 are just rescaled versions of the original probabilities, scaled by a constant
chosen to make the total mass 1.
For a simple example, let A1 be the event that the top card in a well-shuffled standard
deck is a diamond, let A2 be the event that it is a heart, and let B be the event that it
is a red card. Then P (A1) = P (A2) = 1/4 and P (A1|B) = P (A2|B) = 1/2.
10. Fred is working on a major project. In planning the project, two milestones are set up,
with dates by which they should be accomplished. This serves as a way to track Fred’s
progress. Let A1 be the event that Fred completes the first milestone on time, A2 be
the event that he completes the second milestone on time, and A3 be the event that he
completes the project on time.
Suppose that P (Aj+1|Aj ) = 0.8 but P (Aj+1|Ac
behind on his schedule it will be hard for him to get caught up. Also, assume that the
second milestone supersedes the first, in the sense that once we know whether he is
on time in completing the second milestone, it no longer matters what happened with
the first milestone. We can express this by saying that A1 and A3 are conditionally
independent given A2 and they’re also conditionally independent given Ac
(a) Find the probability that Fred will finish the project on time, given that he completes
the first milestone on time. Also find the probability that Fred will finish the project on
time, given that he is late for the first milestone.
(b) Suppose that P (A1) = 0.75. Find the probability that Fred will finish the project
on time.
Solution:
1). To do so, let’s use LOTP to condition on
whether or not A2 occurs:
P (A3|A1) = P (A3|A1, A2)P (A2|A1) + P (A3|A1, A
c c
1
2)P (A2|A ).
Using the conditional independence assumptions, this becomes
P (A3|A2)P (A2|A1) + P (A3|A
c c
1
Similarly,
P (A3|Ac
3 2 2
2)P (A2|A ) = (0.8)(0.8) + (0.3)(0.2) = 0.7.
1) + P (A3|A2)P (A2|A1) = (0.8)(0.3) + (0.3)(0.7) = 0.45.
1) = P (A |A )P (A |A
c c c c
(b) By LOTP and Part (a),
P (A3) = P (A3|A1)P (A1) + P (A3|A
c c
1)P (A1) = (0.7)(0.75) + (0.45)(0.25) = 0.6375.

11. An exit poll in an election is a survey taken of voters just after they have voted. One
major use of exit polls has been so that news organizations can try to figure out as
soon as possible who won the election, before the votes are officially counted. This has
been notoriously inaccurate in various elections, sometimes because of selection bias:
the sample of people who are invited to and agree to participate in the survey may not
be similar enough to the overall population of voters.
Consider an election with two candidates, Candidate A and Candidate B. Every voter
is invited to participate in an exit poll, where they are asked whom they voted for; some
accept and some refuse. For a randomly selected voter, let A be the event that they voted
for A, and W be the event that they are willing to participate in the exit poll. Suppose
that P (W |A) = 0.7 but P (W |Ac
) = 0.3. In the exit poll, 60% of the respondents say
they voted for A (assume that they are all honest), suggesting a comfortable victory for
A. Find P (A), the true proportion of people who voted for A.
Solution: We have P (A|W ) = 0.6 since 60% of the respondents voted for A. Let p =
P (A). Then
P(W|A)P(A) 0.7p
0.6 = P (A|W ) =
P (W |A)P (A) + P (W |Ac)P (Ac)
=
0.7p + 0.3(1 − p)
.
Solving for p, we obtain
P (A) =
9
23
≈ 0.391.
So actually A received fewer than half of the votes!
12. Alice is trying to communicate with Bob, by sending a message (encoded in binary)
across a channel.
(a) Suppose for this part that she sends only one bit (a 0 or 1), with equal probabilities.
If she sends a 0, there is a 5% chance of an error occurring, resulting in Bob receiving a
1; if she sends a 1, there is a 10% chance of an error occurring, resulting in Bob receiving
a 0. Given that Bob receives a 1, what is the probability that Alice actually sent a 1?
(b) To reduce the chance of miscommunication, Alice and Bob decide to use a repetition
code. Again Alice wants to convey a 0 or a 1, but this time she repeats it two more times,
so that she sends 000 to convey 0 and 111 to convey 1. Bob will decode the message by
going with what the majority of the bits were. Assume that the error probabilities are
as in (a), with error events for different bits independent of each other. Given that Bob
receives 110, what is the probability that Alice intended to convey a 1?
Solution:
(a) Let A1 be the event that Alice sent a 1, and B1 be the event that Bob receives a 1.
Then
P(B1|A1)P(A1) (0.9)(0.5)
P (A1|B1) =
P (B |A )P (A ) + P (B |Ac
)P (Ac
)
=
(0.9)(0.5) + (0.05)(0.5)
≈ 0.9474.
1 1 1 1 1 1
(b) Now let A1 be the event that Alice intended to convey a 1, and B110 be the event
that Bob receives 110. Then
P(B110|A1)P(A1)
P (A1|B110) =
P (B |A )P (A ) + P (B |Ac
)P (Ac
)
110 1 1 110 1 1
=
(0.9 ·0.9 ·0.1)(0.5)
(0.9 · 0.9 · 0.1)(0.5) + (0.05 · 0.05 · 0.95)(0.5)
≈ 0.9715.

34
13. Company A has just developed a diagnostic test for a certain disease. The disease
afflicts 1% of the population. As defined in Example 2.3.9, the sensitivity of the test is
the probability of someone testing positive, given that they have the disease, and the
specificity of the test is the probability that of someone testing negative, given that they
don’t have the disease. Assume that, as in Example 2.3.9, the sensitivity and specificity
are both 0.95.
Company B, which is a rival of Company A, offers a competing test for the disease.
Company B claims that their test is faster and less expensive to perform than Company
A’s test, is less painful (Company A’s test requires an incision), and yet has a higher
overall success rate, where overall success rate is defined as the probability that a random
person gets diagnosed correctly.
(a) It turns out that Company B’s test can be described and performed very simply: no
matter who the patient is, diagnose that they do not have the disease. Check whether
Company B’s claim about overall success rates is true.
(b) Explain why Company A’s test may still be useful.
(c) Company A wants to develop a new test such that the overall success rate is higher
than that of Company B’s test. If the sensitivity and specificity are equal, how high
does the sensitivity have to be to achieve their goal? If (amazingly) they can get the
sensitivity equal to 1, how high does the specificity have to be to achieve their goal? If
(amazingly) they can get the specificity equal to 1, how high does the sensitivity have
to be to achieve their goal?
Solution:
(a) For Company B’s test, the probability that a random person in the population is
diagnosed correctly is 0.99, since 99% of the people do not have the disease. For a
random member of the population, let C be the event that Company A’s test yields the
correct result, T be the event of testing positive in Company A’s test, and D be the
event of having the disease. Then
P (C) = P (C|D)P (D) + P (C|D
c
)P (D
c
)
= P (T |D)P (D) + P (T
c
|D
c
)P (D
c
)
= (0.95)(0.01) + (0.95)(0.99)
= 0.95,
which makes sense intuitively since the sensitivity and specificity of Company A’s test
are both 0.95. So Company B is correct about having a higher overall success rate.
(b) Despite the result of (a), Company A’s test may still provide very useful information,
whereas Company B’s test is uninformative. If Fred tests positive on Company A’s test,
Example 2.3.9 shows that his probability of having the disease increases from 0.01 to
0.16 (so it is still fairly unlikely that he has the disease, but it is much more likely than
it was before the test result; further testing may well be advisable). In contrast, Fred’s
probability of having the disease does not change after undergoing Company’s B test,
since the test result is a foregone conclusion.
(c) Let s be the sensitivity and p be the specificity of A’s new test. With notation as in
the solution to (a), we have
P (C) = 0.01s + 0.99p.
If s = p, then P (C) = s, so Company A needs s > 0.99.
If s = 1, then P (C) = 0.01 + 0.99p > 0.99 if p > 98/99 ≈ 0.9899.
If p = 1, then P (C) = 0.01s + 0.99 is automatically greater than 0.99 (unless s = 0, in
which case both companies have tests with sensitivity 0 and specificity 1).

14. Consider the following scenario, from Tversky and Kahneman:
Let A be the event that before the end of next year, Peter will have installed
a burglar alarm system in his home. Let B denote the event that Peter’s
home will be burglarized before the end of next year.
(a) Intuitively, which do you think is bigger, P (A|B) or P (A|Bc
)? Explain your intuition.
(b) Intuitively, which do you think is bigger, P (B|A) or P (B|Ac
)? Explain your intuition.
(c) Show that for any events A and B (with probabilities not equal to 0 or 1), P (A|B) >
P (A|Bc
) is equivalent to P (B|A) > P (B|Ac
).
(d) Tversky and Kahneman report that 131 out of 162 people whom they posed (a)
and (b) to said that P (A|B) > P (A|Bc
) and P (B|A) < P (B|Ac
). What is a plausible
explanation for why this was such a popular opinion despite (c) showing that it is
impossible for these inequalities both to hold?
Solution:
(a) Intuitively, P (A|B) seems larger than P (A|Bc
) since if Peter’s home is burglarized,
he is likely to take increased precautions (such as installing an alarm) against future
attempted burglaries.
(b) Intuitively, P (B|Ac
) seems larger than P (B|A), since presumably having an alarm
system in place deters prospective burglars from attempting a burglary and hampers
their chances of being able to burglarize the home. However, this is in conflict with
(a), according to (c). Alternatively, we could argue that P (B|A) should be larger than
P (B|Ac
), since observing that an alarm system is in place could be evidence that the
neighborhood has frequent burglaries.
(c) First note that P (A|B) > P (A|Bc
) is equivalent to P (A|B) > P (A), since LOTP
says that P (A) = P (A|B)P (B) + P (A|Bc
)P (Bc
) is between P (A|B) and P (A|Bc
) (in
words, P (A) is a weighted average of P (A|B) and P (A|Bc
)). But P (A|B) > P (A) is
equivalent to P (A, B) > P (A)P (B), by definition of conditional probability. Likewise,
P (B|A) > P (B|Ac
) is equivalent to P (B|A) > P (B), which in turn is equivalent to
P (A, B) > P (A)P (B).
(d) It is reasonable to assume that a burglary at his home might cause Peter to install
an alarm system and that having an alarm systems might reduce the chance of a future
burglary. People with inconsistent beliefs about (a) and (b) may be thinking intuitively
in causal terms, interpreting a probability P (D|C) in terms of C causing D. But the
definition of P (D|C) does not invoke causality and does not require C’s occurrence to
precede D’s occurrence or non-occurrence temporally.
15. Let A and B be events with 0 < P (A ∩ B) < P (A) < P (B) < P (A ∪ B) < 1. You are
hoping that both A and B occurred. Which of the following pieces of information would
you be happiest to observe: that A occurred, that B occurred, or that A ∪ B occurred?
Solution: If C is one of the events A, B, A ∪ B, then
P (A ∩ B|C) =
P(A ∩B ∩C)
=
P (C)
P(A ∩B)
.
P (C)
So among the three options for C, P (A ∩ B|C) is maximized when C is the event A.
16. Show that P (A|B) ≤ P (A) implies P (A|Bc
) ≥ P (A), and give an intuitive explanation
of why this makes sense.
Solution: By LOTP,
P (A) = P (A|B)P (B) + P (A|B
c
)P (B
c
).

36
So P (A) is between P (A|B) and P (A|Bc
); it is a weighted average of these two con-
ditional probabilities. To see this in more detail, let x = min(P (A|B), P (A|Bc
)), y =
max(P (A|B), P (A|Bc
)). Then
P (A) ≥ xP (B) + xP (B
c
) = x
and
P (A) ≤ yP (B) + yP (B
c
) = y,
so x ≤ P (A) ≤ y. Therefore, if P (A|B) ≤ P (A), then P (A) ≤ P (A|Bc
).
It makes sense intuitively that B and Bc
should work in opposite directions as evidence
regarding A. If both B and Bc
were evidence in favor of A, then P (A) should have
already reflected this.
17. In deterministic logic, the statement “A implies B” is equivalent to its contrapositive,
“not B implies not A”. In this problem we will consider analogous statements in prob-
ability, the logic of uncertainty. Let A and B be events with probabilities not equal to
0 or 1.
(a) Show that if P (B|A) = 1, then P (Ac
|Bc
) = 1.
Hint: Apply Bayes’ rule and LOTP.
(b) Show however that the result in (a) does not hold in general if = is replaced by ≈.
In particular, find an example where P (B|A) is very close to 1 but P (Ac
|Bc
) is very
close to 0.
Hint: What happens if A and B are independent?
Solution:
(a) Let P (B|A) = 1. Then P (Bc
|A) = 0. So by Bayes’ rule and LOTP,
c c c c c c
P (Ac
|Bc
) =
P(B |A )P (A )
=
P(B |A )P (A )
= 1.
P (Bc
|Ac
)P (Ac
) + P (Bc
|A)P (A) P (Bc
|Ac
)P (Ac
)
(b) For a simple counterexample if = is replaced by ≈ in (a), let A and B be independent
events with P (A) and P (B) both extremely close to 1. For example, this can be done in
the context of flipping a coin 1000 times, where A is an extremely likely (but not certain)
event based on the first 500 tosses and B is an extremely likely (but not certain) event
based on the last 500 tosses. Then P (B|A) = P (B) ≈ 1, but P (Ac
|Bc
) = P (Ac
) ≈ 0.
18. Show that if P (A) = 1, then P (A|B) = 1 for any B with P (B) > 0. Intuitively, this says
that if someone dogmatically believes something with absolute certainty, then no amount
of evidence will change their mind. The principle of avoiding assigning probabilities of
0 or 1 to any event (except for mathematical certainties) was named Cromwell’s rule
by the statistician Dennis Lindley, due to Cromwell saying to the Church of Scotland,
“think it possible you may be mistaken”.
Hint: Write P (B) = P (B ∩ A) + P (B ∩ Ac
), and then show that P (B ∩ Ac
) = 0.
Solution: Let P (A) = 1. Then P (B ∩ Ac
) ≤ P (Ac
) = 0 since B ∩ Ac
⊆ Ac
, which shows
that P (B ∩ Ac
) = 0. So
P (B) = P (B ∩ A) + P (B ∩ A
c
) = P (A ∩ B).
Thus,
P(A ∩B) P(A ∩B)
P (A|B) = = = 1.
P (B) P (A ∩ B)

1
1
19. Explain the following Sherlock Holmes saying in terms of conditional probability, care-
fully distinguishing between prior and posterior probabilities: “It is an old maxim of
mine that when you have excluded the impossible, whatever remains, however improb-
able, must be the truth.”
Solution: Let E be the observed evidence after a crime has taken place, and let
A1, A2, . . . , An be an exhaustive list of events, any one of which (if it occurred) would
serve as an explanation of how the crime occurred. Assuming that the list A1, . . . , An
exhausts all possible explanations for the crime, we have
P (A1 ∪ A2 ∪ · · · ∪ An|E) = 1.
Sherlock’s maxim says that
P (An|E, A
c c c
1) = 1,
1, A1, . . . , An−
i.e., if we have determined that all explanations other than An can be ruled out, then the
remaining explanation, An, must be the truth, even if P (An) and P (An|E) are small.
To prove Sherlock’s maxim, note that
P (A
c
, . . . , A
c
|E) = P (A
c
, . . . , A
c
, A
c
|E) + P (A
c
, . . . , A
c
, A |E),
1 n−1 1 n−1 n 1 n−1 n
where the first term on the right-hand side is 0 by De Morgan’s laws. So
P(Ac
,Ac
,...,Ac
,An|E)
P (An|E, Ac c c
1) = 1 1 n−1
= 1.
1, A1, . . . , An−
P (Ac c c
1, A1, . . . , An−1|E)
20. The Jack of Spades (with cider), Jack of Hearts (with tarts), Queen of Spades (with a
wink), and Queen of Hearts (without tarts) are taken from a deck of cards. These four
cards are shuffled, and then two are dealt.
(a) Find the probability that both of these two cards are queens, given that the first
card dealt is a queen.
(b) Find the probability that both are queens, given that at least one is a queen.
(c) Find the probability that both are queens, given that one is the Queen of Hearts.
Solution:
(a) Let Qi be the event that the ith card dealt is a queen, for i = 1, 2. Then P (Qi) = 1/2
since the ith card dealt is equally likely to be any of the cards. Also,
1 1 1
P (Q1, Q2) = P (Q1)P (Q2|Q1) =
2
·
3
=
6
.
As a check, note that by the naive definition of probability,
1 1
Thus,
P (Q1, Q2) = 4
=
6
.
2
P(Q1 ∩Q2) 6 1
P (Q1 ∩ Q2|Q1) = = = .
P (Q ) 1
3
1 2
(b) Continuing as in (a),
P(Q1 ∩Q2) P(Q1 ∩Q2) 6 1
P (Q1∩Q2|Q1∪Q2) =
P (Q
=
∪ Q ) P (Q ) + P (Q ) − P (Q
=
∩ Q ) 1
+ 1 1 =
5
.
1 2 1 2 1 2 2 2
− 6
Another way to see this is to note that there are 6 possible 2-card hands, all equally

38
|
likely, of which 1 (the “double-jack pebble”) is eliminated by our conditioning; then by
definition of conditional probability, we are left with 5 “pebbles” of equal mass.
(c) Let Hi be the event that the ith card dealt is a heart, for i = 1, 2. Then
P(Q1 ∩H1 ∩Q2) + P(Q1 ∩Q2 ∩H2)
P (Q1 ∩ Q2|(Q1 ∩ H1) ∪ (Q2 ∩ H2)) =
1
P (Q1
1 1
∩ H1) + P (Q2
1
∩ H2)
= 4
·3
+ 4
·3
1 1
=
1
,
3
4
+ 4
using the fact that Q1 ∩ H1 and Q2 ∩ H2 are disjoint. Alternatively, note that the
conditioning reduces the sample space down to 3 possibilities, which are equally likely,
and 1 of the 3 has both cards queens.
21. A fair coin is flipped 3 times. The toss results are recorded on separate slips of paper
(writing “H” if Heads and “T” if Tails), and the 3 slips of paper are thrown into a hat.
(a) Find the probability that all 3 tosses landed Heads, given that at least 2 were Heads.
(b) Two of the slips of paper are randomly drawn from the hat, and both show the
letter H. Given this information, what is the probability that all 3 tosses landed Heads?
Solution:
(a) Let A be the event that all 3 tosses landed Heads, and B be the event that at least
2 landed Heads. Then
P (A, B) P (A) 1/8 1
P (A|B) = = = = .
P (B) P (2 or 3 Heads) 4/8 4
(b) Let C be the event that the two randomly chosen slips of paper show Heads. Then
P(C|A)P (A)
P (A|C) =
P (C)
=
P(C|A)P(A)
P (C|A)P (A) + P (C|2 Heads)P (2 Heads) + P (C|1 or 0 Heads)P (1 or 0 Heads)
1
= 8
1 1 3 1
8
+ 3
· 8
+ 0 · 2
=
1
.
2
Alternatively, let Ai be the event that the ith toss was Heads. Note that
P (A) 1/8 1
P (A|Ai, Aj ) =
P (A , A )
=
1/4
=
2
i j
for any i = j. Since this probability is 1/2 regardless of which 2 slips of paper were
drawn, conditioning on which 2 slips were drawn gives
P (A C) =
1
.
2
22. s A bag contains one marble which is either green or blue, with equal probabilities. A
green marble is put in the bag (so there are 2 marbles now), and then a random marble
is taken out. The marble taken out is green. What is the probability that the remaining
marble is also green?
Solution: Let A be the event that the initial marble is green, B be the event that the

removed marble is green, and C be the event that the remaining marble is green. We
need to find P (C|B). There are several ways to find this; one natural way is to condition
on whether the initial marble is green:
P (C|B) = P (C|B, A)P (A|B) + P (C|B, A
c
)P (A
c
|B) = 1P (A|B) + 0P (A
c
|B).
To find P (A|B), use Bayes’ rule:
P(B|A)P(A) 1/2 1/2 2
P (A|B) =
So P (C|B) = 2/3.
= = = .
P (B) P (B|A)P (A) + P (B|Ac)P (Ac) 1/2 + 1/4 3
Historical note: This problem was first posed by Lewis Carroll in 1893.
23. s Let G be the event that a certain individual is guilty of a certain robbery. In gathering
evidence, it is learned that an event E1 occurred, and a little later it is also learned that
another event E2 also occurred. Is it possible that individually, these pieces of evidence
increase the chance of guilt (so P (G|E1) > P (G) and P (G|E2) > P (G)), but together
they decrease the chance of guilt (so P (G|E1, E2) < P (G))?
Solution: Yes, this is possible. In fact, it is possible to have two events which separately
provide evidence in favor of G, yet which together preclude G! For example, suppose
that the crime was committed between 1 pm and 3 pm on a certain day. Let E1 be
the event that the suspect was at a specific nearby coffeeshop from 1 pm to 2 pm that
day, and let E2 be the event that the suspect was at the nearby coffeeshop from 2 pm
to 3 pm that day. Then P (G|E1) > P (G), P (G|E2) > P (G) (assuming that being in
the vicinity helps show that the suspect had the opportunity to commit the crime), yet
P (G|E1 ∩ E2) < P (G) (as being in the coffeehouse from 1 pm to 3 pm gives the suspect
an alibi for the full time).
24. Is it possible to have events A1, A2, B, C with P (A1|B) > P (A1|C) and P (A2|B) >
P (A2|C), yet P (A1 ∪ A2|B) < P (A1 ∪ A2|C)? If so, find an example (with a “story”
interpreting the events, as well as giving specific numbers); otherwise, show that it is
impossible for this phenomenon to happen.
Solution: Yes, this is possible. First note that P (A1 ∪ A2|B) = P (A1|B) + P (A2|B) −
P (A1 ∩ A2|B), so it is not possible if A1 and A2 are disjoint, and that it is crucial to
consider the intersection. So let’s choose examples where P (A1 ∩ A2|B) is much larger
than P (A1 ∩ A2|C), to offset the other inequalities.
Story 1 : Consider two basketball players, one of whom is randomly chosen to shoot two
free throws. The first player is very streaky, and always either makes both or misses both
free throws, with probability 0.8 of making both (this is an extreme example chosen for
simplicity, but we could also make it so the player has good days (on which there is a
high chance of making both shots) and bad days (on which there is a high chance of
missing both shots) without requiring always making both or missing both). The second
player’s free throws go in with probability 0.7, independently. Define the events as Aj :
the jth free throw goes in; B: the free throw shooter is the first player; C = Bc
. Then
P (A1|B) = P (A2|B) = P (A1 ∩ A2|B) = P (A1 ∪ A2|B) = 0.8,
P (A1|C) = P (A2|C) = 0.7, P (A1 ∩ A2|C) = 0.49, P (A1 ∪ A2|C) = 2 · 0.7 − 0.49 = 0.91.
Story 2 : Suppose that you can either take Good Class or Other Class, but not both. If
you take Good Class, you’ll attend lecture 70% of the time, and you will understand the
material if and only if you attend lecture. If you take Other Class, you’ll attend lecture
40% of the time and understand the material 40% of the time, but because the class is
so poorly taught, the only way you understand the material is by studying on your own

40
j ) for j ∈ {1, 2}. Assume that 10%
and not attending lecture. Defining the events as A1: attend lecture; A2: understand
material; B: take Good Class; C: take Other Class,
P (A1|B) = P (A2|B) = P (A1 ∩ A2|B) = P (A1 ∪ A2|B) = 0.7,
P (A1|C) = P (A2|C) = 0.4, P (A1 ∩ A2|C) = 0, P (A1 ∪ A2|C) = 2 · 0.4 = 0.8.
25. s A crime is committed by one of two suspects, A and B. Initially, there is equal
evidence against both of them. In further investigation at the crime scene, it is found
that the guilty party had a blood type found in 10% of the population. Suspect A does
match this blood type, whereas the blood type of Suspect B is unknown.
(a) Given this new information, what is the probability that A is the guilty party?
(b) Given this new information, what is the probability that B’s blood type matches
that found at the crime scene?
Solution:
(a) Let M be the event that A’s blood type matches the guilty party’s and for brevity,
write A for “A is guilty” and B for “B is guilty”. By Bayes’ rule,
P(M|A)P(A) 1/2 10
P (A|M ) =
P (M |A)P (A) + P (M |B)P (B)
=
1/2 + (1/10)(1/2)
=
11
.
(We have P (M |B) = 1/10 since, given that B is guilty, the probability that A’s blood
type matches the guilty party’s is the same probability as for the general population.)
(b) Let C be the event that B’s blood type matches, and condition on whether B is
guilty. This gives
1 10 1 2
P (C|M ) = P (C|M, A)P (A|M ) + P (C|M, B)P (B|M ) =
10
·
11
+
11
=
11
.
26. s To battle against spam, Bob installs two anti-spam programs. An email arrives,
which is either legitimate (event L) or spam (event Lc
), and which program j marks as
legitimate (event Mj ) or marks as spam (event M c
of Bob’s email is legitimate and that the two programs are each “90% accurate” in the
sense that P (Mj |L) = P (M c c
j |L ) = 9/10. Also assume that given whether an email is
spam, the two programs’ outputs are conditionally independent.
(a) Find the probability that the email is legitimate, given that the 1st program marks
it as legitimate (simplify).
(b) Find the probability that the email is legitimate, given that both programs mark it
as legitimate (simplify).
(c) Bob runs the 1st program and M1 occurs. He updates his probabilities and then
runs the 2nd program. Let P̃(A) = P (A|M1) be the updated probability function after
running the 1st program. Explain briefly in words whether or not P̃(L|M2) = P (L|M1 ∩
M2): is conditioning on M1 ∩M2 in one step equivalent to first conditioning on M1, then
updating probabilities, and then conditioning on M2?
Solution:
(a) By Bayes’ rule,
9 1
P(M1|L)P(L) 10
·10 1
P (L|M1) = =
P (M ) 9
1
1 9 =
2
.
1
10
· 10
+ 10
· 10

10
9
(b) By Bayes’ rule,
P(M1, M2|L)P(L) ( 9
)
2 1
· 10
P (L|M1, M2) = =
P (M , M )
= .
10
1 2 ( 9
)2 · 1
+ ( 1
)2 · 9
10 10 10 10
(c) Yes, they are the same, since Bayes’ rule is coherent. The probability of an event
given various pieces of evidence does not depend on the order in which the pieces of
evidence are incorporated into the updated probabilities.
27. Suppose that there are 5 blood types in the population, named type 1 through type 5,
with probabilities p1, p2, . . . , p5. A crime was committed by two individuals. A suspect,
who has blood type 1, has prior probability p of being guilty. At the crime scene blood
evidence is collected, which shows that one of the criminals has type 1 and the other
has type 2.
Find the posterior probability that the suspect is guilty, given the evidence. Does the
evidence make it more likely or less likely that the suspect is guilty, or does this depend
on the values of the parameters p, p1, . . . , p5? If it depends, give a simple criterion for
when the evidence makes it more likely that the suspect is guilty.
Solution: Let B be the event that the criminals have blood types 1 and 2 and G be the
event that the suspect is guilty, so P (G) = p. Then
P(B|G)P(G) p2p p
P (G|B) =
P (B|G)P (G) + P (B|Gc)P (Gc)
=
p p + 2p p (1 − p)
=
p + 2p (1 − p)
,
2 1 2 1
since given G, event B occurs if and only if the other criminal has blood type 2, while
given Gc
, the probability is p1p2 that the elder criminal and the younger criminal have
blood types 1 and 2 respectively, and also is p1p2 for the other way around.
Note that p2 canceled out and p3, p4, p5 are irrelevant. If p1 = 1/2, then P (G|B) =
P (G). If p1 < 1/2, then P (G|B) > P (G), which means that the evidence increases the
probability of guilt. But if p1 > 1/2, then P (G|B) < P (G), so the evidence decreases
the probability of guilt, even though the evidence includes finding blood at the scene of
the crime that matches the suspect’s blood type!
28. Fred has just tested positive for a certain disease.
(a) Given this information, find the posterior odds that he has the disease, in terms of
the prior odds, the sensitivity of the test, and the specificity of the test.
(b) Not surprisingly, Fred is much more interested in P (have disease|test positive),
known as the positive predictive value, than in the sensitivity P (test positive|have disease).
A handy rule of thumb in biostatistics and epidemiology is as follows:
For a rare disease and a reasonably good test, specificity matters much more than sen-
sitivity in determining the positive predictive value.
Explain intuitively why this rule of thumb works. For this part you can make up some
specific numbers and interpret probabilities in a frequentist way as proportions in a
large population, e.g., assume the disease afflicts 1% of a population of 10000 people
and then consider various possibilities for the sensitivity and specificity.
Solution:
(a) Let D be the event that Fred has the disease, and T be the event that he tests
positive. Let sens = P (T |D), spec = P (T c
|Dc
) be the sensitivity and specificity (re-
spectively). By the odds form of Bayes’ rule (or using Bayes’ rule in the numerator and
the denominator), the posterior odds of having the disease are
P(D|T)
=
P(D) P(T|D) sens
= (prior odds of D) .

P (Dc
|T ) P (Dc
) P (T |Dc
) 1 − spec

42
sens + q
(1 − spec)
.
(b) Let p be the prior probability of having the disease and q = 1 − p. Let PPV be the
positive predictive value. By (a) or directly using Bayes’ rule, we have
PPV =
sens
p
For a concrete example to build intuition, let p = 0.01 and take sens = spec = 0.9 as
a baseline. Then PPV ≈ 0.083. In the calculations below, we describe what happens
if sensitivity is changed while specificity is held constant at 0.9 or vice versa. If we
can improve the sensitivity to 0.95, the PPV improves slightly, to 0.088. But if we can
improve the specificity to 0.95, the PPV improves to 0.15, a much bigger improvement.
If we can improve the sensitivity to 0.99, the PPV improves to 0.091, but the other way
around the PPV improves drastically more, to 0.48. Even in the extreme case that we
can make the sensitivity 1, the PPV only improves to 0.092. But in the extreme case
that we can make the specificity 1, the PPV becomes 1, the best value possible!
To further the intuitive picture, imagine a population of 10000 people, in which 1% (i.e.,
100 people) have the disease. Again take sens = spec = 0.9 as a baseline. On average,
there will be 90 true positives (correctly diagnosed diseased people), 10 false negatives
(misdiagnosed diseased people), 8910 true negatives (correctly diagnosed healthy peo-
ple), and 990 false positives (misdiagnosed healthy people). This is illustrated in the
figure below (not to scale).
90
true positives
990
100 false positives
10
false negatives
10000 people
healthy
9900 people
test -
8910 people
true negatives
The PPV corresponds to the number of true positives over the number of positives,
which is 90/(90+990) ≈ 0.083 in this example. Increasing specificity could dramatically
decrease the number of false positives, replacing 990 by a much lower number; on the
other hand, increasing sensitivity could at best increase the number of true positives
from 90 to 100 here.
29. A family has two children. Let C be a characteristic that a child can have, and assume
that each child has characteristic C with probability p, independently of each other and
of gender. For example, C could be the characteristic “born in winter” as in Example
2.2.7. Show that the probability that both children are girls given that at least one is a
girl with characteristic C is 2−p
, which is 1/3 if p = 1 (agreeing with the first part of
4−p
Example 2.2.5) and approaches 1/2 from below as p → 0 (agreeing with Example 2.2.7).
Solution: Let G be the event that both children are girls, A be the event that at least

Random documents with unrelated
content Scribd suggests to you:

SOUTH-EAST APSE.
they would not have been
undertaken. There were
conflagrations in 1170 and 1271, and
in the fearful riots of 1272 the
cathedral was set on fire by the
citizens. Still, when the presbytery
was repaired in 1362, it seems to
have been roofed again in wood. In
1463 the wooden spire was struck by
lightning, and set fire to the roofs
both of nave and presbytery. At last
the monks had to bestir themselves.
To secure the spire against fire they
rebuilt it in stone instead of wood;
and, to make the nave and presbytery
fireproof, they made up their minds to
vault both in stone. Between 1463
and 1472 Bishop Lyhart put up over the nave the present
magnificent lierne vault, and at his death bequeathed two thousand
marks to his successor to continue the work. Bishop Goldwell vaulted
the presbytery between 1472 and 1499. It seems to have been very
difficult to get the funds for this costly work. Bishop Goldwell,
however, was a personal friend of the Pope, who had consecrated
him with his own hands; and he had not much difficulty in
persuading the Pope to grant a perpetual indulgence in the terms
that “all who came to the cathedral on Trinity Sunday and Lady Day,
and made an offering towards the fabric, should be entitled to an
indulgence of twelve years and forty days.” The transepts had still
wooden roofs. It required another fire—in 1509—in which these
roofs were consumed, to compel the monks to complete the vaulting
of the cathedral. This was done in the time of Bishop Nix. At the end
of four hundred years Norwich cathedral was at length fireproof.
And did they do nothing merely for prettiness’ sake? Well, here, as
at Gloucester, they set to work to do what was quite unnecessary—
to harmonise the Norman ground-story of the presbytery with the

CHOIR.
clerestory of 1631. By one of those marvellous pieces of
engineering, of which the mediæval architects were so fond—we
saw a conspicuous example at Carlisle—while retaining the Norman
triforium and the Perpendicular clerestory above, they managed to
remodel the Norman piers on either side of the presbytery, and to
take out the semicircular arches bodily and replace them by the
fashionable arch of the period—a depressed four-centered arch. This
was done by Bishop Goldwell—no doubt before he put up the vault
above (c. 1475).
The only other great work was
the rebuilding of the cloisters,
also forced on the monks by a
great fire—that of 1272. This
work was executed exceedingly
slowly, the window tracery
ranging from Geometrical,
through Curvilinear, to
Perpendicular work.
One word more about the
superb interior. It is hardly too
much to say that the interior of
this cathedral—but second-rate in
point of dimensions—is
unequalled in all England. One
reason is that it is vaulted
throughout. Ely, Peterborough St. Albans, Rochester, Romsey,
Waltham, Southwell—with their paltry wooden ceilings—are not to
be compared for a moment with Norwich. Gloucester and Chichester
naves are vaulted, but the vaults are too slight and flimsy for the
stern and massive work below. Durham vault is strong and
satisfactory. But the lierne vault of Norwich is a far more glorious
crown and finish than the rude work of Durham. It might be thought
that the richness and magnificence of the lierne vault of Norwich
would be out of harmony with the simplicity and heaviness of
Norman piers and triforium and clerestory. It is not so. A tower, like

that of Magdalen College, Oxford, may be ever so plain below, and
yet terminate fitly with a glorious coronal of battlements, parapet
and pinnacles. So it is with this interior.
Its rivals are to be found in Winchester and Tewkesbury. But at
Winchester the vaults of nave and presbytery are cut in two by the
unvaulted transept. Norwich and Tewkesbury are vaulted everywhere
—from east to west and from north to south. And in both, the vaults
being uniform in character, and not changing character half-way as
at Gloucester, weld together the spreading limbs of the church into a
marvellous unity.
But there is another fine feature about the interior of Norwich, as
in that of Gloucester: it is the striking contrast of light and shade, of
shadowy nave and brilliant choir. Hereford presents us with the
reverse effect—bright nave and gloomy choir. Both effects are
dramatic; both, doubtless, are unintentional. If they had known how,
or could have afforded it, the Hereford people would have flooded
their choir with sunshine, Norwich and Gloucester their naves. The
mediæval builders wanted none of these dramatic contrasts of light
and shade; they were always working to get rid of the dim religious
light that nowadays we venerate; they would have liked their
churches lighted thoroughly well throughout. What they wanted was
the light, uniformly good, of Lichfield and Exeter: or Salisbury, bright
and gay as a ball-room.
But the most subtle and most important element in the beauty of
the interior of Norwich is to be found in its proportions. The nave is
of an immense length, but it is very narrow. York, Canterbury,
Lincoln, Durham, all have naves far too short for their breadth. And
what is more important still is the ratio of the height of Norwich nave
to its span. In most English cathedrals it is 2 to 1; but in Norwich
nave the ratio rises to 2⁴⁄₇, and in the presbytery it is 3 to 1. In
Norwich presbytery, then, we have just those proportions which we
find in the great Gothic cathedrals of France, but in England hardly
anywhere except in Westminster Abbey. People admire Norwich

APSE OF CHOIR.
presbytery and Westminster
presbytery for the same reason, and,
no doubt, in most cases, without
knowing what the reason is.
One thing more remains to be said
in praise of Norwich, as of Gloucester.
It is that all the glory of the church is
concentrated at one spot, and that
the most important spot in the
church. It is in approaching the high
altar that vaults and clerestory soar
aloft, that loveliest vistas open out
into ambulatory and chapels, while
the noble windows above fill all with
light and atmosphere. “I would back
it,” says Dean Goulburn, “against any
similar effect in almost any cathedral in Christendom.”

“A
The Cathedral Church of Christ,
Oxford.
FROM THE FIELDS.
bout the year of our Lord 727, there lived in Oxford a Saxon
prince named Didan, who had an only child, Frideswide (‘bond
of peace’). Seeing that he had large possessions and inheritances,
and that she was likely to enjoy most of them after his decease,
Frideswide told her father that he could not do better than bestow
them upon some religious fabric where she and her spiritual sisters
might spend their days in prayers and in singing psalms and hymns
to God. Wherefore the good old man built a church, and committed
it wholly to the use of his daughter, purposely to exercise her
devotion therein; and other edifices adjoining to the church, to serve

as lodging-rooms for Frideswide and twelve virgins of noble
extraction. There she became famous for her piety and for those
excellent parts that nature had endowed her withal; and Algar, King
of Leicester, became her adorer by way of marriage. Finding that he
could not prevail with her by all the entreaties and gifts imaginable,
he departed home, but sent to her ambassadors with this special
and sovereign caution, that if she did not concede, to watch their
opportunity and carry her away by force. Frideswide was inexorable.
Wherefore at the dawning of the day the ambassadors clambered
the fences of the house, and by degrees approaching her private
lodging, promised to themselves nothing but surety of their prize.
But she, awakening suddenly and discovering them, and finding it
vain to make an escape, being so closely besieged, fervently prayed
to the Almighty that He would preserve her from the violence of
those wicked persons, and that He would show some special token
of revenge upon them for this their bold attempt. Wherefore the
ambassadors were miraculously struck blind, and like madmen ran
headlong yelling about the city. But Algar was filled with rage, and
intended for Oxford, breathing out nothing but fire and sword.
Which thing being told to Frideswide in a dream, with her sisters the
nuns Katherine and Cicely, she fled to the riverside, where there
awaited her a young man with a beautiful countenance and clothed
in white, who, mitigating their fear with pleasant speech, rowed
them up the river to a wood ten miles distant. There the nuns
sheltered in a hut, which ivy and other sprouts quickly overgrew,
hiding them from sight of man. Three years Frideswide lived in
Benton wood, when she came back to Binsey and afterwards to
Oxford, in which place this maiden, having gained the triumph of her
virginity, worked many miracles; and when her days were over and
her Spouse called her, she there died.” Such is the account of her
which Anthony-a-Wood drew from William of Malmesbury and Prior
Philip of Oxford, both of whom unfortunately lived long after the
events which they narrate.
I. In the east walls of the north choir-aisle and the Lady chapel
three small rude arches have recently been found, and outside, in

NAVE.
the gardens, the foundations of the walls of three apses. Hence it
has been concluded that we have here the eastern termination of
Frideswide’s eighth-century church. It may be so, but the central
arch seems very small for the chancel-arch of an aisled church. It is
indeed a foot wider than the chancel-arch of the Saxon church at
Bradford-on-Avon, but that tiny church has no aisles. Moreover, if the
side-arches led into aisles, they would be likely to be of the same
height, whereas the southern arch is considerably the higher of the
two.
II. At some later period—
perhaps in the eighth or ninth
century—the foundation was
converted into one of secular
canons, married priests taking
the place of the nuns (cf. Ely).
The secular canons themselves in
turn gave way to monks, and
these in 1111 to regular canons—
i.e., canons living in monastic
fashion under the rule (regula) of
St. Augustine, as at Bristol,
Ripon, and Carlisle.
The first business, probably, of
the secular canons was to house
themselves—i.e., to build
themselves the usual cloister, with its appanages of chapter-house,
refectory, dormitory, etc. Of the chapter-house which they built, c.
1125, the doorway still remains.
In 1004 King Ethelred had rebuilt the Saxon church; and probably
it was found possible to put this church into such repair as would
allow the services to be held in it for the time being. At any rate, it
was not till 1158 that they commenced the present cathedral, which
they finished in 1180, leaving not a stone standing of Ethelred’s
cathedral. Of the theory that the present cathedral is in the main the

one built in 1004, I would prefer to say nothing, had it not been
adopted in a recent history of the cathedral; suffice to say that, like
the sister theory that Waltham Abbey was built in 1060, it is an
absolute impossibility. The hands of the archæological clock cannot
be turned 160 years back in this preposterous fashion.
The twelfth-century church was very remarkable in plan. Not only
had it an aisled nave and an aisled choir, but it had the architectural
luxury, unparalleled in our Norman architecture except in the vast
churches of Winchester and Ely, of eastern and western aisles to its
transepts. The site, however, was cramped to the south, and so the
southern transept was shorter than the northern one; moreover, this
short transept later on lost its west aisle, which was lopped off to
allow the cloister to be extended. For the same reason—lack of room
—the slype, or vaulted passage, which in all monastic institutions
connected the cloister with the cemetery, instead of being built
between the transept and the cloister, was built inside the church, as
at Hexham, curtailing still further the floor area of the north
transept. It was therefore because the church was so cramped to
the south, that the other transept was given aisles on both sides.
Instead of an eastern aisle, the south transept had merely a square
chapel projecting eastward.
But the canons wanted also a Lady chapel, for the church seems
to have been dedicated originally to the Holy Trinity, St. Mary and St.
Frideswide. The normal position of a Lady chapel was to the east of
the sanctuary. But here also the canons were cramped; for quite
close to the east end of the church ran the city wall. To get in a Lady
chapel, therefore, they had to build an additional aisle north of the
north aisle of the choir. This was three bays long. It was probably
walled off from the transept, but opened into the north choir-aisle by
three Norman arches, reconstructed later on. The same arrangement
is found at Ripon. There was also a short chapel projecting eastward
from the northernmost bay of the east aisle of the north transept.
The east end, as at Rochester and Ripon, was square. The present
east end is a fine composition by Scott, more or less conjectural. The

work commenced, as usual, at the east, as is shown by the gradual
improvement westward in the design of the capitals. The evidence of
the vaulting, too, points in the same direction. In the choir-aisle the
ribs are massive and heavy; in the western aisle of the north
transept they are lighter; in the south aisle of the nave they are
pointed and filleted.
The transepts are narrower than the nave and choir; the tower,
therefore, is oblong, and, as at Bolton Priory, its narrow sides have
pointed arches: semicircular arches would have been too low. The
faces of the piers of the towers are flat, because the stalls of the
canons were placed against them and in the eastern bays of the
nave, leaving the whole eastern limb as sanctuary.
The clerestory walls are only 41½ feet high; therefore, to have
adopted the usual Norman design—viz., triforium on the top of pier-
arcade—would have made the interior look very squat: so, instead of
building the triforium above the pier-arcade, it was built beneath it.
The lofty pier-arches, thus gained, add greatly to the apparent
height and dignity of the interior. The lower arches, however, which
carry the vault of the aisle behind, are corbelled into the piers in
very clumsy fashion. The design is not original; it was worked out at
Romsey in a single bay of the nave, but, being thought ugly, was
promptly abandoned. It is worked out more successfully in
Dunstable Priory church and Jedburgh Abbey. The clerestory
windows of the nave would be built not much before 1180; naturally,
therefore, they are pointed. The capitals of all the twelfth-century
work are full of interest. Indeed, Transitional capitals—each an
experiment, and all differing—partly conventional, partly naturalistic,
with a dash of Classic—are to me much more interesting than any of
the Gothic capitals, except perhaps the naturalistic capitals of the
later Geometrical period. There is a great sameness about the
foliated capitals of the Early English, Curvilinear, and Perpendicular
periods. I need hardly say that no one of these capitals came from
Ethelred’s church.

The whole church is exceedingly interesting. It fills a niche in the
history of English architecture all by itself. It is not the plain and
austere Transitional work of the Cistercians. On the other hand, it
has not yet the lightness and grace of Ripon; still less the charm of
Canterbury choir, Chichester presbytery, Wells and Abbey Dore—
Gothic in all but name. In spite of its foliated capitals, in spite of a
pointed arch here and there, it is a Romanesque design; yet not so
Romanesque as Fountains, Kirkstall, Furness.
III. In the Lancet period (1190-1245) the works went on apace. An
upper stage was added to the tower, and on that the spire was built
—the first large stone spire in England. It is a Broach spire: i.e., the
cardinal sides of the spire are built right out to the eaves, so that
there is no parapet. On the other hand, instead of having broaches
at the angle, it has pinnacles. Moreover, to bring down the thrusts
more vertically, heavy dormer-windows are inserted at the foot of
each of the cardinal sides of the spire: altogether a very logical and
scientific piece of engineering, much more common in the early
spires of Northern France than in England.
The chapter-house also was rebuilt (c. 1240); rectangular, to fit
the cloister. Also, the canons rebuilt both the Lady chapel and the
adjoining transeptal chapel. Lancet work will be seen in all the piers
on the south side of the Lady chapel, and in the second and third
piers from the west, on its north side. The cult of the Virgin, much
fostered by the Pope, Innocent III., was at its height in the
thirteenth century. The Lady chapels of Bristol, Hereford, Salisbury,
Winchester and Norwich were contemporaries of that of Oxford.
IV. To the latter half of the Geometrical period belong the
fragments of the pedestal of St. Frideswide’s shrine, which has
beautiful naturalistic foliage like that of the contemporary pedestal of
St. Thomas of Hereford, a.d. 1289. Some twenty years later is the
fine canopied tomb of Prior Sutton.
V. In the Curvilinear period (1315-1360) the eastern chapel of the
north transept was pulled down, and in its place was built a chapel
of four bays, with four side windows of singularly beautiful tracery,

CHOIR.
and all different. They contain
fourteenth-century glass, which
should be compared with that in St.
Lucy’s chapel and in Merton College
chapel. The bosses are very beautiful:
one of them has a representation of
the water-lilies of the adjacent
Cherwell. Hard by is the tomb of Lady
Montacute, who gave the canons
about half the Christ Church meadows
to found a chantry. The chapel goes
by various names: St. Katharine’s
chapel, the Latin chapel, and the
Divinity chapel. It contains good
poppy-heads of Cardinal Wolsey’s
time.
About the same time the eastern chapel of the south transept—St.
Lucy’s chapel—was enlarged. The tracery of its east window starts in
an unusual fashion below the spring of the arch.
Also the Norman windows were replaced here and there by large
windows with flowing tracery, to improve the lighting of the church.
VI. There is little to show for the long Perpendicular period (1360-
1485), except the insertion of a few large Perpendicular windows,
and the so-called “Watching-chamber,” the lower part of which is the
tomb of a merchant and his wife, the upper part probably, the
chantry belonging to it, c. 1480.
VII. In the Tudor period, however, the canons were exceedingly
busy. They set to work to make the whole church fireproof by
covering choir, transepts, and nave with stone vaults. The choir vault
is rather overdone with prettinesses. It is a copy—and an inferior
one—of the massive vault of the Divinity School, which was
completed c. 1478. Canon Zouch, who died in 1503, left money to
proceed with the vault of the north transept, beneath which is his
tomb. Only a small portion of this was completed. In the clerestory

of the nave also corbels were inserted to support a stone vault; but
the resources of the canons seem to have failed, and the rest of the
church received roofs of wood. Another considerable work was the
rebuilding of the cloisters.
VIII. Finally, the whole establishment was granted in 1524 to
Cardinal Wolsey, who pulled down the three western bays of the
nave, as obstructing his new quadrangle: one bay has been recently
rebuilt.
IX. In 1542 Henry VIII. founded the new diocese of Oxford. Till
1546 the seat of the bishopric was at Osney Abbey. On the
suppression of the abbey it was transferred to Wolsey’s confiscated
foundation; and the ancient Priory church became a cathedral, while
at the same time it is the chapel of the college of Christ Church.
There is an interesting contemporary window in the south choir
aisle, showing the first bishop of Oxford, King, with Osney Abbey on
one side. The “merry Christ Church bells” came from the tower
shown in this window.
X. At the entrance to the Great Hall is the last bit of good Gothic
done in England, a sort of chapter-house in fan-tracery.
XI. The cathedral possesses a charming Jacobean pulpit, and a
large amount of fine Flemish glass of the seventeenth century—all of
it taken out and stowed away in some lumber-room at a recent
restoration, except one window at the west end of the north aisle of
the nave, in order to insert some sham mediæval windows.
XII. There are also five windows from designs by Sir Edward
Burne-Jones—three of them of great beauty; good windows by
Clayton and Bell in the end walls of the transepts; and a charming
reredos by Mr. Bodley, who also has the credit of the bell tower.

S
The Cathedral Church of St. Peter,
Peterborough.
FROM THE SOUTH.
t. Augustine landed in Kent a.d. 597. In the next year Peada and
Wolfhere, successive kings of Mercia, founded a monastery at
Peterborough, then called Medeshamstead (“the homestead in the
meadow”), and consecrated the church in the names of St. Peter, St.
Paul, and St. Andrew. Then said King Wulfhere with a loud voice:
“This day do I freely give to St. Peter and to the abbot and to the
monks of this monastery these lands and waters and meres and fens
and weirs; neither shall tribute or tax be taken therefrom. Moreover
I do make this monastery free, that it be subject to Rome alone; and
I will that all who may not be able to journey to Rome should repair
hither to St. Peter.” This consecration took place in 664. In 870 this,
the first church, was destroyed by the Danes. It was not fully rebuilt
till 972. Abbot Elsinus (1006-1055) collected many curios: pieces of
the swaddling clothes, of the manger of the cross, and of the

sepulchre of Christ; of the garments of the Virgin, of Aaron’s rod, a
bone of one of the Innocents, bits of St. John the Baptist, St. Peter,
and St. Paul, the body of St. Florentinus, for which he gave 100 lbs.
of silver, and, most precious of all, the incorruptible arm of the
Northumbrian king, Oswald, believed by half the population of
England to be an effectual cure for diseases which defied the
material power of drugs. Here is Bede’s account of it: “When Oswald
was once sitting at dinner with Bishop Aidan, on the holy day of
Easter, and a silver dish of dainties was before him, the servant,
whom he had appointed to relieve the poor, came in on a sudden,
and told the king that a great multitude of needy persons were
sitting in the streets begging alms of the king. He immediately
ordered the meat set before him to be carried to the poor, and the
dish also to be cut in pieces and divided among them. At which sight
the Bishop laid hold of the King’s right hand, and said, ‘May this
hand never perish,’ which fell out according to his prayer; for his arm
and hand being cut off from his body, when he was slain in battle,
remain entire and incorrupted to this day, and are kept in a silver
case as revered relics in St. Peter’s church in the royal city.” Even
King Stephen came to see it; and, what is more, remitted to the
monks forty marks which they owed him. Benedict was a monk at
Canterbury when Becket was murdered; and when he became Abbot
of Peterborough in 1177, he brought with him the slabs of the
pavement which were stained with the blood of the martyr,
fragments of his shirt and surplice, and two vases of his blood. So
that the monastery was called “Peterborough the Proud,” and waxed
rich and mighty, and church and close were holy ground, and all
pilgrims, even though of royal blood, put off their shoes before
passing through the western gateway of the close.
I. The second Saxon church of 972 seems to have lasted till 1116,
when it was destroyed by fire, and the present church, the third,
was commenced. The foundations of part of this Saxon cathedral
have been recently disinterred beneath the present south transept.
It was cruciform, with a square east end. The east limb was 23 feet
each way; the transept was 88 feet long. Its walls were under three

CHOIR AND TRANSEPT.
feet thick, so that it cannot have been intended for a vault. There is
no proof that the nave was ever built.
II. In 1116 the Saxon cathedral was seriously injured by a great
fire, and next year Abbot John of Sais (Seez) commenced the
present Norman cathedral. In 1140 the monks entered on the new
choir, which was now complete, together with the eastern aisles and
eastern wall of the transepts. It is possible that the monks patched
up the damaged Saxon church sufficiently to allow service to be held
in it from 1116 to 1140.
When they entered into their new Norman choir, the first thing
they did, probably, was to pull down the choir and transept of the
Saxon church, and on the site to erect the rest of the present south
transept.
Then they built the rest of the
north transept. It will be noticed
that it is superior in design to the
south transept, its windows are
splayed, and their ornamentation
of later character. This north
transept is illustrated by M.
Viollet-le-Duc as a specially fine
example of English Romanesque.
Next would be built the
remaining piers and arches of the
crossing, and a low lantern tower
of one story only. But the western
piers would not stand without
abutment, and so a certain
amount of the eastern bays of
the nave must have been built at the same time. This comprised two
bays of the triforium, for the tympana of the two eastern bays of the
triforium have rude ornaments not found elsewhere in the nave.
Below, it probably comprised four piers and four arches, for the four

NAVE.
eastern piers on the north side have different bases from those to
the west.
Hitherto the north wall of the Saxon
nave, if built, may have been retained
to shut in the cloister on the north.
Now it would be pulled down and
replaced by the wall of the present
south aisle of the nave. Then would
come the wall of the north aisle, and
finally the pier-arcade and triforium,
but not yet the clerestory of the nave.
The nave was to be in plan precisely
like that of Durham: it was to be a
short nave; the central aisle to have
eight bays; the side aisles were to
have only seven bays, the end of each
aisle being occupied by a tower, as at
Durham. The ground stories of these
towers now form the third bays from
the west on either side of the nave. It will be noticed that the third
piers from the west are exceptionally massive and strong, and that
in this bay the aisle-walls are thickened. The wide arches of these
bays were intended to open up the towers into the nave.
But the towers were not built. The Ely monks over the way were
building a nave with no less than twelve bays, and with a western
transept as well. The Peterborough monks would not like to be
outdone by Ely; so they determined also to have a long nave and a
western transept as well. They built only ten bays to the twelve of
Ely; on the other hand, their nave, excluding western transept, was
211 feet long, while that of Ely was only 208 feet. About the same
time, or probably a little earlier, the clerestory of the nave—in which
pointed arches occur—was built. All this work may be assigned to
Abbot Benedict (1175-1193), who is said by Swapham and John to
have built the whole nave as far as—but not including—the present
west front. The statements of Swapham, however, must be wrong

VAULTING UNDER SOUTH-WEST
TOWER.
here. He was still living c. 1240;
so that he was only a boy when
the nave was finished. He may
possibly in his boyhood have seen
the clerestory of the nave built,
and, in writing half a century
after, have thought that Benedict
who built the clerestory, had built
the triforium and ground-story
also. But the documentary
evidence at Peterborough must
be received with the utmost
scepticism. All that we know for
certain is that the choir and the
eastern portions of the central
transepts were built between
1118 and 1140; and that the
central transept, central tower,
nave and western transept were built between 1140 and 1190.
III. Lancet (1190-1245).—The east end of the church consisted of
three parallel apses. The apses of the aisles were now replaced by
narrow oblong bays: those next to the New Building.

WEST FRONT.
In the middle of the Lancet period was erected the grand façade
in front of the Transitional western transept. It is not so much a
façade, however, as an open portico or piazza. Several interesting
engineering problems were involved. One was, how to keep up the
three gigantic arches. If they had spread to north or south, the
whole façade would have collapsed. To prevent their spreading,
therefore, flanking towers were built to north and south; which in
later days were weighted with spires. But there was a more serious
danger. The two great isolated piers might be pushed outwards by
the western thrust of the arches of the nave. These thrusts the
builders stopped by building two towers; one over the westernmost
bay of each aisle of the nave. The northern of these towers was
soon after heightened; the other—the Bell Tower—remains low. The
central gable had to be narrow, because it is the termination of the
nave roof. The side-arches and side-gables had to be wide, to span
the space from the nave to the sides of the Transitional façade
behind. Though much narrower, however, the central gable rises as

GABLE.
high as the lateral gables, being made to spring at a higher level;
and it is made to look as important as the broad side-gables by
being given the company of two powerful pinnacles. Thus the main
features of this magnificent design are due to difficulties of planning
and construction. The design is said to be drawn from Lincoln; it is
more likely that it is an amplification of John de Cella’s lovely design
for the west front of St. Albans. Abbot Acharius, who may well have
commenced the work (1200-1210) had been Prior of St. Albans
under John de Cella. Judging from the billet and nebula ornament on
the gables, and from the arcading, in which semicircular arches and
round-headed trefoils occur, the façade was designed in the very
beginning of the thirteenth century.
The west front of Peterborough has
been severely criticised, especially by
Mr. Pugin. To many it will ever seem
the highest effort of English art, and
to be at once the most original and
most successful façade either in
English or in Continental Gothic. Yet,
magnificent and poetic as it is, we
have not the full effect contemplated
by the mediæval builders. They meant
to have four towers, not three. The
north-west tower was once crowned
by a wooden spire; we may be sure
that there would have been a spire
also on the south-west tower. Add,
too, in the background, the tall spire
which was to be added to the central
tower, and you have a group before which even Lichfield and Lincoln
would pale into insignificance. But, even curtailed as it is, the design
attains the sublime. When first its Titanic arches rose into the blue
sky, its builders may well have repeated the psalmist’s words: “Lift
up your heads, O ye gates; and be ye lift up, ye everlasting doors;

FROM SOUTH-EAST.
and the King of Glory shall come in.” They had built a worthy portal
to the House of the Almighty.
IV. Geometrical (1245-1315).—In this period the bell tower was
carried up; and a magnificent Lady chapel was built (c. 1290), like
that at Bristol, to the north of the choir, but detached from it. It
could not be built east of the choir, as a high road passed close to
the apse. This Lady chapel was pulled down in the seventeenth
century for the sake of its materials.
V. Curvilinear.—In this period
the weight of the Norman tower,
which had of course very thick
walls, and was three or four
stories high, was found to be too
much for the exceptionally weak
piers on which it stood. Warned,
perhaps, by the fate of the
central towers of Ely and Wells,
both of which collapsed about
this time, they took down the
Norman tower, and built a new
one (which has recently been
rebuilt), much lighter and much
lower. And they strengthened its
eastern and western semicircular
arches by inserting pointed
arches beneath them. The south-west spire was also built—a design
of exquisite beauty.
VI. Perpendicular.—The monks wanted to have a Galilee porch, and
they inserted one between the piers of the west front, where it was
constructionally useful by keeping the piers from bulging in. The
wooden screens were now inserted in the central transept.
Peterborough, after 1116, seems to have had a singular immunity
from fire; so, very unlike Norwich, the monks did not take the
slightest trouble to make their church fireproof. The whole of the

high roofs are of wood. That of the nave may possibly be the
original twelfth-century ceiling. A twelfth-century wooden roof still
covers the Bishop’s Palace at Hereford. The choir has a wooden vault
of the fifteenth century.
VIII. In another respect the history of the church is uneventful.
The eastern limb must have been exceedingly inconvenient, for
there was no processional aisle or ambulatory round the apse. Every
other large church pulled down or altered its eastern limb to suit the
ritual: the Peterborough monks, always conservative and always
behind the times, did not provide a processional aisle till the latter
days of Gothic. And even then they took a very long time about it.
The works seems to have been suspended in 1471, and not resumed
till 1496. Even then, good conservatives that they were, they did not
pull down the apse, but erected the New Building round it. It is a
rich specimen of Tudor work, with a fan vault.
IX. In the matter, too, of the roof-drainage the Peterborough
monks were slow to move. Instead of dripping eaves they
constructed gutters and parapets to the aisles in the early years of
the thirteenth century, and to the apse a little later. It was not till c.
1330 that they provided the high roofs of nave and choir with
gutters and parapets; and, with their wonted conservatism, they
retained the Norman corbel-table.
X. What the monks cared most about was the lighting of the
church. This they were always trying to improve. In the thirteenth
century they inserted large geometrical windows in the western
transept, and c. 1290 others in the aisles of the central transepts to
light the altars placed there. Moreover, the Norman windows in the
aisles of the nave were replaced by wide windows of five lights. In
the Curvilinear period the triforium windows were transformed, and
charming flowing tracery, with rear-arches, was inserted in the
windows of the apse, which then looked into the open air, but now
look into the New Building. In the Perpendicular period some
seventy-five windows were either enlarged or filled with rectilinear
tracery. The builders certainly achieved their object. The cathedral is

RETRO-CHOIR
well lighted. We may be thankful that
they did not stick a great
Perpendicular window in each end of
the central transept.
XI. In 1541 the church was made a
cathedral on the new foundation.
Henry VIII. is said to have preserved
it as a mausoleum to his first wife,
Catharine of Arragon, who is buried in
the choir. It is wretchedly built—the
west front and the New Building as
badly as the Norman work—and
practically without foundations. Much
underpinning has been done, and
more is required. The west front has
been saved for the present by
judicious treatment.

R
The Cathedral Church of St. Peter
and St. Wilfrid, Ripon.
ipon minster has passed through strange vicissitudes. It was
founded c. 660 as a monasterium or minster for Scottish monks
attached to the Celtic church. Soon afterwards it was taken away
from them and granted to the famous St. Wilfrid. In 678 the church
became a cathedral, but only during the lifetime of Bishop Eadhed.
Ultimately it passed into the hands of regular canons of the
Augustinian Order. It was dissolved with the other collegiate
churches by Edward VI. It was made collegiate once more by James
I., but with dean and prebendaries instead of Augustinian canons. In
1836, for the second time, it became a cathedral.
I. Both the minsters built by St. Wilfrid—Ripon and Hexham—
retain their crypts. He was a Romaniser in architecture as in ritual,
and well acquainted with Italy. So his seventh-century church at
Ripon was modelled after the early Christian basilicas which he had
seen at Rome. Like them, it had a confessionary or crypt, which still

exists, beneath the central tower; like them, it was orientated to the
west. He seems even to have brought over Italian masons to direct
or to execute the work, for the crypt is vaulted, and the vaulting is of
excellent construction; the masonry is smooth, and is covered “with
a fine and very hard plaster which takes a polish.” At its west end
was the altar, at its east end an aperture through which a glimpse of
the interior might be obtained from the Saxon nave. Round the walls
are little niches in which lights were placed. “St. Wilfrid’s Needle” is
merely a niche with the back knocked through. Similar Saxon crypts
remain at Hexham and Wing, and a Norman crypt at St. Peter-in-
the-East, Oxford. They usually consisted of a small central chamber,
with a passage all round it. There were two staircases descending
from either side of the nave; pilgrims went down one flight of steps,
proceeded along the passage, getting a glimpse of the relics through
openings in the wall of the central chamber, and then returned up
the other flight of steps into the nave.
IV. Norman.—Early in the twelfth century a Norman cathedral
seems to have been built, wholly or in part, by Archbishop Thurstan.
Of this there remains only an apsidal building, with crypt beneath,
on the south side of the south aisle of the present choir. An
eleventh-century chapel formerly existed, with crypt beneath it, in
precisely the same situation at Worcester; there is a twelfth-century
chapel in the same position in Oxford cathedral. In Oxford this
chapel was the Lady chapel. It may be that the Ripon chapel also
may have been a Lady chapel. For if the Norman choir was of the
same length eastward as at present, it would have been impossible
to build a Lady chapel of the type of that, the crypt of which still
exists at Winchester, to the east of the choir; the ground falls far too
steeply eastward. Moreover, the so-called Lady loft now existing
would seem, from its name, to be merely an upper story added to a
Lady chapel. This Norman chapel formerly opened into the Norman
church; traces of the arches may be seen in the walls. In the
buttress is a curious room which may have been a sacristy, a
lavatory, a prison, or an anchorite’s cell, like the one in the east end
of Ludlow church.

NAVE.
V. Transitional.—From 1154 to
1181 there ruled at York a man of
the greatest energy and power—
Archbishop Roger. He condemned
his two Norman churches at York
and Ripon; made no attempt, as
at Peterborough and Ely, to
improve them; simply pulled
them down, and started again de
novo. The two new minsters
seem to have been somewhat
similar: both had square east
ends, both had exceptionally
broad naves. But Ripon minster
was merely the church of
Augustinian canons, therefore it
was not planned in cathedral
fashion. Our ancient collegiate and parish churches seem to have
followed some other model than the early Christian basilicas which
furnished the plans of the cathedrals. Most of our parish churches
originally were without aisles; and even large churches of the regular
canons frequently had no aisles to the nave. This was the case in
Roger’s new church at Ripon, and at Bolton Priory (also
Augustinian). Later on, indeed, the Ripon canons added north and
south aisles to their nave, and the Bolton canons a north aisle to
their nave—they could not add a south aisle also, as they had a
cloister to the south. But originally both churches had aisleless
naves. That of Ripon was 40 feet broad—broader than any nave in
England, except York, which is of the breadth of 45 feet. The
combination of unaisled nave and aisled choir must have produced a
very remarkable interior; quite unlike anything now existing in
England, but to be paralleled by the Spanish cathedral at Gerona
(illustrated in Street’s “Gothic Architecture of Spain”).
Of this Transitional nave nothing is now left except two fragments,
one at the east, and one at the west end on either side. All the rest

has been replaced by sixteenth-century piers, arches, and clerestory.
But if in imagination the two ends of the Transitional nave are joined
together—it is well to do so in an actual drawing—the design of the
whole of the original nave can be recovered with exactitude. A very
remarkable design it was. It consisted of three stories; the lower
story was simply a blank wall. The second, the triforium, was merely
a passage in the thickness of the wall, ornamented in front by a tall
pointed arcade. The clerestory had three tall slender lancet windows
in each bay, all of the same height, separated by two detached piers.
The strangest feature of the nave was that neither in the ground-
story nor in the triforium were there any windows. Everywhere else
people were trying to get all the windows possible into their
churches; here alone a “dim, religious light” was preferred. And
filtering in, as it did, through small lancet windows at a great height,
as in Pugin’s cathedral at Killarney, the effect must have been most
dramatic. The destruction of this unique nave is one of the heaviest
losses that English architecture has sustained.
Of the central tower, the south-east pier has been rebuilt; the
north-east and south-west piers have been cased. The north and
west arches of the tower survive; the south and east arches have
been rebuilt. The nave was considerably wider than the central aisle
of the choir; the tower was therefore not built square; the northern
arch being set obliquely, and not parallel to the southern one.
Outside, however, the north side of the tower is corbelled out till the
tower becomes square.
The design of the choir is best seen in the east side of the north
transept, which retains the original round-headed windows. In the
choir the western windows of the north aisle were converted into
lancets in the thirteenth century. The design of transept and choir is
almost Greek in its severity. Very effective is the contrast of broad
wall-surface and plain splayed window with the light and slender
shafted arcades of triforium and clerestory. In proportions, too, it is
superior to nearly all later designs. The pier-arches are tall and
narrow, and the triforium thoroughly subordinated to the tall
clerestory; the proportions approximate closely to those of

NORTH TRANSEPT.
Westminster Abbey and Beverley
Minster. It is remarkable, too, for the
studied absence of foliated ornament.
Not that the builders could not design
a foliated capital; they have left one
or two, in unnoticed corners of the
north transept, to show their powers.
All the capitals of the choir are
moulded capitals, as at Roche Abbey
—perhaps occurring here for the first
time. Being a first attempt, they can
hardly be considered a success; they
were soon to be improved upon in the
French crypt of Canterbury. The
designer relied on architectural effects
pure and simple, and was followed in
his ascetic self-restraint sixty years
later in the eastern transept of Fountains and at Salisbury. Even
more remarkable is the complete abolition of Norman ornament. The
billet, the zigzag, the whole barbaric congeries of Norman ornament
is contemptuously cast aside. In this respect, indeed, Ripon is much
more advanced than Canterbury choir, which was not commenced till
1174. The clerestory, however, is of a familiar Norman type, being an
adaptation of that of Romsey and Waltham Abbey, and Peterborough
and Oxford cathedrals; it was reproduced a little later in Hexham
choir.
The vaulting-shafts rest on the abaci, French fashion. In the choir
there are five vaulting-shafts, which in the clerestory diminish to
one. The effect is not satisfactory, and a different treatment is
adopted in the transept. It is noteworthy that the vaulting of the
north transept aisle contains an exceptionally early example of ridge-
ribs, both longitudinal and transverse.
In spite of round-headed windows and round-headed arches here
and there, the whole design of the interior is light and graceful,
thoroughly Gothic. Externally it is just the reverse; but for a pointed

arcade in the clerestory one might imagine one was back again in
the early days of the twelfth century. But when one compares the
interior with that of Oxford Cathedral, which is precisely
contemporary, and in which the spirit of Gothic is wholly absent,
suspicion rises to certainty: Ripon Minster must have been designed
under French influence. The tall, acute, pointed pier-arches of Notre
Dame, Chalons-sur-Marne, and Sens (commenced c. 1140) reappear
in Ripon choir, and the undoubtedly French choir of Canterbury. The
Chalons triforium reappears in the north wall of Ripon transept; the
Chalons clerestory in that of Ripon nave. The clerestory of Ripon
choir is practically that of the French choir of Canterbury, itself
probably suggested by that of Sens. French too, are the vaulting-
shafts of the choir, insecurely balancing on the abaci; and the
broadness and plain splays of the windows.
IV. Lancet.—To this period belong the vaulting and piers of the
present chapter-house; and the west front, which, like York transept
and Southwell choir, is attributed to Archbishop Gray (ob. 1255). The
west front is too flat; deficient in play of light and shade; correct and
uninteresting. It is ruined by the loss of its wooden spires, removed
in 1664; and by the miserable little pinnacles put up in 1797. Before
the aisles were built, these towers projected clear of the nave, their
inner walls are Transitional; but the Transitional arch has been taken
out and replaced by one of the Lancet period.
V. Geometrical.—The lower row of lancets in the west front once
had charming tracery, inserted early in the Geometrical period. This
was destroyed by Scott.
About 1280 the east end of the choir seems to have collapsed—
partly, perhaps, in consequence of the steep fall of the ground
eastwards. It was rebuilt, with the damaged portions of the choir,
with exceptional strength in consequence. The east end is a
vigorous, massive design, something like that of Guisborough or
Selby. Only the eastern portion of the choir has flying buttresses.
The clerestory windows have an inner arcade. Ripon choir alone, of

CHOIR, LOOKING EAST.
English cathedrals,
possesses a glazed
triforium, the lean-to roof of
the aisles having been
replaced by a flat roof (see
Ely).
VI. To the Curvilinear
period (1315-1360) belong
the Lady loft and the
sedilia. The latter originally
stood one bay more to the
west. In the Tudor period
the arches of the sedilia
seem to have received the
present clumsy shafts.
VII. Perpendicular (1360-1485).—In 1458 the southern and eastern
sides of the central tower collapsed, greatly damaging the adjacent
parts of the choir and transept, as well as the stalls. The eastern
aisle of the south transept and much of the south side of the choir,
as well as part of the tower, had to be rebuilt; and about 1490 the
present choir-stalls were put up. In the choir the builders, with a
conservatism which does them credit, both in the work of 1280 and
in that of 1458, preserved all they could of the twelfth-century work,
and both in the Geometrical and the Perpendicular bays of the
triforium retained the semicircular arch of the older design. The
result is a curious blend of styles. Starting from the east, the first
pier on the north side is Geometrical, the rest Transitional. On the
south side the first pier is Geometrical, the second Transitional, the
third and fourth Perpendicular. To give more support to the tower,
the north-east and south-west piers were cased; the south-east pier
was rebuilt, and the southern and eastern arches were rebuilt. To
strengthen the eastern piers of the tower, the two western bays of
the arcade of the choir were blocked up, and a massive choir-screen
was inserted c. 1480.

CHOIR, LOOKING WEST.
VIII. Tudor.—Early in the
sixteenth century the canons
unhappily determined to give
their unique church more of the
look of a cathedral by adding
aisles to the nave. It is pleasant
to add that they were
unsuccessful. The nave is
exceedingly low in proportion to
its exceptional span, and being,
moreover, unprovided with a
triforium, does not look in the
least like a cathedral, but like a
very inferior parish church.
Externally, the buttresses are of
fine composition, and if the pinnacles were completed, the nave
would be very handsome externally.
In 1593 the central spire—of timber and lead—was struck by
lightning, and in 1660 it was removed. It was 120 feet high. In 1664,
for fear of a similar catastrophe, the western spires also were
removed. The result is that, seen from a distance, minus spires and
minus pinnacles, Ripon Minster is stunted and squat.

R
The Cathedral Church of St. Andrew,
Rochester.
FROM NORTH-EAST.
ochester and London, next to Canterbury, are the oldest of all
the English bishoprics, unless, indeed, we are prepared to accept
a pre-Augustine bishopric of Hereford. St. Augustine, soon after his
landing in 597, came to preach at Rochester. His reception was not
encouraging; the rude people hung fish-tails to his coat. Wherefore
in anger the saint prayed “that the Lord would smite them in

posteriora to their everlasting ignominy. So that not only on their
own but on their successors’ persons similar tails grew ever after.”
The worst of it was that the story spread, and not only Rochester
people but all English folk were believed on the Continent to be
caudati (tailed). So that even in the sixteenth century “an
Englishman now cannot travel in another land by way of
merchandise or any other honest occupying, but it is most
contumeliously thrown in his teeth that all Englishmen have tails.”
Among St. Augustine’s Italian missioners were St. Justus and St.
Paulinus. St. Justus became first bishop of Rochester in 604. St.
Paulinus, after eight years of mission work in Northumbria, became
bishop of Rochester in 633. The first English bishop was St. Ythamar
(644-655). These three were the chief local saints of Rochester in
early days.
St. Augustine and his missioners had come from the monastery of
St. Andrew, Rome. To St. Andrew, therefore, they dedicated the first
Saxon cathedral. In 1542 the cathedral was re-dedicated to Christ
and the Blessed Virgin Mary of Rochester. Till 1077 the cathedral was
served by secular canons; Gundulph replaced them by Benedictine
monks.
I. In 1888 the foundations of an early church were found. It had
an apse, but neither aisles nor transepts; walls only 2 feet 4 inches
thick; 42 feet long, 28 feet broad. From the resemblance of its plan
to that of St. Pancras, Canterbury, and the presence of Roman brick
in the walls, it seems as likely to be a Romano-British as an Anglo
Saxon church.
II. Between 1080 and 1089 Bishop Gundulph completed a Norman
cathedral, except the western part of the nave. In plan it was
entirely different from any Norman cathedral of the day: one can
hardly help believing that it must have been designed by an
Englishman. The plan of it is given in the Builder. It was a long
oblong aisled cathedral, with nave and aisles running on without a
break from the west end of the nave to the east end of the choir. But
in the choir the side-aisles were cut off from the central aisle by a

solid wall, as in the contemporary choir of St. Albans and in the
Premonstratensian abbey of St. Radigund, near Dover. There was
probably no crossing, and therefore no central tower. There were no
transepts proper; but, as in such Anglo-Saxon churches as Worth,
low porch-like transepts projected north and south, with a breadth of
only 14 feet. The east end, as in most Anglo-Saxon churches, was
square; and there projected from it eastward a small square chapel.
Beneath was a crypt, the western part of which remains. There were
two towers, both abnormal in position. The southern tower was set
in the angle of the choir and the south transept, and may have been
the belfry. The other tower, fragments of which still remain, was set
in a similar position, but entirely detached. Being detached, and
having walls six feet thick, it was no doubt a military keep. Gundulph
was fond of building keeps; those of the Tower of London and
Malling still exist. Rochester was exposed to and had suffered from
attacks of the Danes, sailing up the Medway, in 840, 884, and 999.
There was a striking memento of them on the great west doors of
the cathedral, which Pepys, as late as 1661, found “covered with the
skins of Danes.” We may conjecture that it was as a refuge against
similar attacks that Gundulph built the northern keep.
All this work of Gundulph’s is now gone except portions of the
crypt, the keep, and the nave. The original monastery was built in
the normal position, south of the nave. To enclose the cloister,
therefore, on the north, the south side of the nave was proceeded
with next. The south aisle-wall is very thin—as was customary in
Anglo-Saxon architecture—and we may conjecture that English
influence stopped at this point; for the piers and arches of the nave
are quite Norman in character. Of Gundulph’s nave there remain on
the south side five arches, together with the lower parts of the walls
of both aisles. It is very doubtful whether he built any part of the
triforium or clerestory. At present his work can only be seen in its
original condition from the side of the aisles. The pier arches had
originally two square orders, which remain unaltered on the side of
the aisle (cf. Winchester transept). Gundulph’s masonry was in rough
tufa.

Welcome to our website – the perfect destination for book lovers and
knowledge seekers. We believe that every book holds a new world,
offering opportunities for learning, discovery, and personal growth.
That’s why we are dedicated to bringing you a diverse collection of
books, ranging from classic literature and specialized publications to
self-development guides and children's books.
More than just a book-buying platform, we strive to be a bridge
connecting you with timeless cultural and intellectual values. With an
elegant, user-friendly interface and a smart search system, you can
quickly find the books that best suit your interests. Additionally,
our special promotions and home delivery services help you save time
and fully enjoy the joy of reading.
Join us on a journey of knowledge exploration, passion nurturing, and
personal growth every day!
testbankmall.com

Introduction to Probability 1st Blitzstein Solution Manual

More Related Content

Similar to Introduction to Probability 1st Blitzstein Solution Manual (20)

Recently uploaded (20)

Introduction to Probability 1st Blitzstein Solution Manual