Chapter1 draft

1
After studying this chapter you should know
• when a binomial distribution is a suitable model for a
practical situation
• how to calculate probabilities using a binomial
distribution
• how to use tables to ﬁnd cumulative binomial
probabilities.

Binomial distribution

From rolling dice to quality control on
a manufacturing production line, the
Binomial Distribution has proved to be
a very useful model to describe a wide
range of problems.

1

CHAPTER 1

1.1 You can use factorial notation to find the number of arrangements of some
objects.

In S1 you saw how to list the number of different arrangements of some objects.

Example 1

Find all possible arrangements of
a 3 objects where one is red, one is blue and one is green,
b 4 objects where 2 are red and 2 are blue.

Suppose the first one is red, then there
a The first object can be chosen in 3 ways. are 2 possibilities for the second. If
The second object can be chosen in 2 the first was blue then there are still 2
possibilities for the second. So the number
ways. of ways should be multiplied.
The third object can be chosen in only
1 way.
Here there are 3 ϫ 2 ϫ 1 ϭ 6 possible 3 ϫ 2 ϫ 1 can be written as 3! or 3
factorial.
arrangements.

These can be listed as follows:
Try and list the possible ways in a
RBG BRG GRB systematic manner. For example, suppose
RGB BGR GBR the first is R then …

b If the red objects are labelled R1 and R2
and the blue objects B1 and B2, then you
can treat the objects as 4 different
ones and there are 4! ways of arranging
them. However, arrangements with
R1R2 … are identical to arrangements
with R2R1 … and so the total number
needs to be divided by 2. Dividing by 2 is the same as dividing by
2!
A similar argument applies to the Bs and so
4!
Number of arrangements is: ______ ϭ 6 You divide by:
2! ϫ 2! 2! for the 2 reds and 2! for the 2 blues.
These can be listed as follows:
RRBB RBBR RBRB Try and work systematically when listing.
For example here the second row is
BBRR BRRB BRBR simply the first with R swapped for B.

2


᭿ n different objects can be arranged in
n! ‫ ؍‬n ؋ (n ؊ 1) ؋ (n ؊ 2) … 3 ؋ 2 ؋ 1 ways.
This is sometimes written as
᭿ n objects with a of one type and (n ؊ a) of another n
( a ) or nCa. You met this
n!
can be arranged in ____________ ways. formula in C2 in connection
a! ؋ (n ؊ a)! with the binomial theorem.

Example 2

A child is playing with some identically shaped coloured bricks. Find the number of different
arrangements that can be made from
a 6 red and 3 blue bricks, b 5 red and 7 blue bricks.

(6 ϩ 3)! Use the above formula with n ϭ 9
a There are ________ ways
6!3! and a ϭ 3.

ϭ 9ϫ8ϫ7ϫ6ϫ…ϫ1
______________________
Notice that 6! is a factor of 9!
3ϫ2ϫ1ϫ6ϫ…ϫ1

ϭ 9 ϫ 8 ϫ 7 ϭ 84
_________
6
Notice that 9 ϭ 9 since if a ϭ 3
( ) ( )
This can simply be calculated as 9 or 9
( ) ( ) 6 3
6 3 then n Ϫ a ϭ 6 and vice versa.
on a calculator.
(7 ϩ 5)!
b There are _______ or 12 ways
( ) Use the ( n ) formula with n ϭ 12
7! ϫ 5! 7 a
ϭ 792 ways and a ϭ 7

The formula for the number of arrangements can be helpful in probability questions.

Example 3

A fair die is rolled 8 times. Find the probability of:
a no sixes, b only 3 sixes, c four twos and 4 sixes.

Let S ϭ the outcome ‘the die lands on a six’. So SЈ is the outcome that the die
does not land on a six.

a P(no sixes) ϭ P(SЈ SЈ SЈ SЈ SЈ SЈ SЈ SЈ) There is only one arrangement of 8
rolls each having the outcome SЈ.
8
5
6( )
ϭ __ ϭ 0.23256 … P(S) ϭ _ and so P(SЈ) ϭ 1 Ϫ _ ϭ _.
1
6
1
6
5
6

ϭ 0.233 (3 s.f.) Final answers should be left as exact
fractions or decimals to 3 s.f.

3

CHAPTER 1

( _ )3 for the 3 sixes.
1

b P(3 sixes) ϭ P(3S and 5SЈ in any order) 6

( _ )5 for the 5 non-sixes.
5
6

1 3 5 5 8!
( )( )
ϭ __ __ ϫ ____
6 6 3!5! 8!
There are 8 or 8 ϭ ____
( ) ( )
ϭ 0.10419 … 3 5 5!3!
arrangements of 3S and 5SЈ.
ϭ 0.104 (3 s.f.)

c Let T ϭ the outcome ‘the die lands on a two’. There are 4S and 4T and
P(4T and 4S in any order) P(S) ϭ P(T ) ϭ _.
1
6

4 4
1 1 8!
6( ) ( __ )
ϭ __
6
ϫ ____
4!4! ( _ )4 for 4T and ( _ )4 for 4S.
1
6
1
6

ϭ 0.00004167 … 8!
There are 8 ϭ ____
( )
ϭ 0.0000417 (3 s.f.) 4 4!4!
arrangements of 4S and 4T.

Exercise 1A

1 A large bag contains counters of different colours.
Find the number of arrangements for the following selections
a 5 counters all of different colours,
b 5 counters where 3 are red and 2 are blue,
c 7 counters where 2 are red and 5 are green,
d 10 counters where 4 are blue and 6 are yellow,
e 20 counters where 2 are yellow and 18 are black.

2 A bag contains 4 red, 3 green and 8 yellow beads.
Five beads are selected at random from the bag without replacement.
Find the probability that they are
a 5 yellow beads,
b 2 red and 3 yellow,
c 4 red and 1 green.

3 A fair die is rolled 7 times.
Find the probability of getting
a no fives,
b only 3 fives,
c 4 fives and 3 sixes.
4


1.2 You can use the binomial theorem to find probabilities.

Example 4

A biased coin has probability p of landing heads when it is thrown.
The coin is thrown 10 times.
Find, in terms of p, the probability of the coin landing heads
a 10 times, b 7 times, c 4 times.

There is only 1
Let the random variable X ϭ number of times (out of a possible arrangement of 10
10) the coin lands heads. heads.

a P(X ϭ 10) ϭ p ϫ p ϫ p ϫ p ϫ p ϫ p ϫ p ϫ p ϫ p ϫ p ϭ p10 Assuming each
throw is independent
b If the coin does not land heads you can multiply
the probabilities
it must land tails. together.
Let q ϭ probability of landing tails
There are only 2
possibilities, H or T.
P(X ϭ 7) ϭ 10 p7q3
( )
7
qϭ1Ϫp
ϭ 120p7q3

c P(X ϭ 4) ϭ 10 p4q6
( ) ( 10 ) ϭ 120 for the number of arrangements
7
4
of 7H and 3T. Then p7 for the 7 heads and q3
ϭ 210p q 4 6
for the 3 tails.

( 10 ) ϭ 210 for the number of arrangements of
4
4H and 6T. Then p4 for the 4H and q6 for the 6T.

You met the binomial expansion in C2. Consider the binomial expansion of:

( p ϩ q)10 ϭ p10 ϩ 10 p9q ϩ 10 p8q2 ϩ 10 p7q3 ϩ … ϩ 10 p4q6 ϩ … ϩ q10
( ) ( ) ( ) ( )
9 8 7 4

You should notice that the probabilities in Example 4 are terms in this binomial expansion. You
could use the full binomial expansion to write down the probability distribution for the random
variable X.

x 0 1 2 3 4 5 6 7 8 9 10

P(X ‫ ؍‬x) q10 … … … ( 10) p q
4
4 6
… … ( 10) p q
7
7 3
… … p10

Notice that the power of p is the value of x.

5

CHAPTER 1

1.3 You need to know when a binomial distribution is a suitable model.

If you look again at Example 4 you can see that there are 4 conditions required for a binomial
distribution.

᭿ A ﬁxed number of trials, n. The coin was thrown 10 times, so n ϭ 10.

᭿ Each trial should be success or failure. The coin could land H or T on each throw.

The distribution is called binomial because
there are only 2 cases for each trial.

You use this when you multiply the
᭿ The trials are independent. probabilities together.

᭿ The probability of success, p, at each trial is You assumed that the probability of a coin
constant. landing heads was p for each throw.

᭿ If these conditions are satisﬁed we say that the random variable X(‫ ؍‬the number of
successes in n trials) has a binomial distribution and write
‘B’ for binomial
X ~ B(n, p)
n for the number of trials
p for the probability of success at each trial.

and P(X ‫ ؍‬x) ‫ ( ؍‬n )px(1 ؊ p)n ؊ x
x This formula is in the formula book.

Sometimes n is called the index and p the parameter of the binomial distribution.

Example 5

The random variable X ~ B(12, _)
1
6

Find a P(X ϭ 2) b P(X ϭ 9) c P(X р 1)

2 10 2 10
1 5 12! 1 5
a P(X ϭ 2) ϭ 12
2 ( ) ( __ ) ( __ )
6 6 ( ) ( __ )
ϭ ____ __
2!10! 6 6
Use the formula with n ϭ 12,
p ϭ _ and x ϭ 2.
1
6

ϭ 0.29609 …
ϭ 0.296 (3 s.f.)
9 3
1 5 Use the formula with n ϭ 12,
b P(X ϭ 9) ϭ 12
9 ( ) ( __ ) ( __ )
6 6 p ϭ _ and x ϭ 9.
1
6

ϭ 0.00001263 …
ϭ 0.0000126 (3 s.f.)

6


Interpret the inequality to identify
c P(X р 1) ϭ P(X ϭ 0) ϩ P(X ϭ 1) the values of X and then use the
12 1 11
formula.
5 1 5
( )
ϭ __
6 ( ) ( __ ) ( __ )
ϩ 12
1 6 6
ϭ 0.112156 … ϩ 0.26917 …
ϭ 0.38133 …
ϭ 0.381 (3 s.f.)

Example 6

Explain whether or not a binomial distribution can be used to model the following situations.
In cases where it can be used, give a deﬁnition of the random variable and suggest suitable values
for n and p.
a The number of throws of a die until a six is obtained.
b The number of girls in a family of 4 children.
c The number of red balls selected when 3 balls are drawn from an urn which contains 15 white
and 5 red balls.
In questions of this type consider
carefully whether any of the
four conditions for a binomial
a Binomial is not suitable since the number of
distribution are not satisfied.
throws (trials) is not fixed. Remember all four conditions must
hold for a binomial.
b Number of trials is fixed (n ϭ 4).
Each trial is success or failure. A ‘trial’ is having a child.
Trials are probably independent. Having a girl is a ‘success’, a boy is
The probability of having a girl should ‘failure’.
be constant.
The gender of the first child
So X ϭ number of girls in families of 4 shouldn’t influence the gender of
children and X ~ B(4, 0.5) the second. (Identical twins may be
a problem here.)
c Let R ϭ number of red balls selected.
The number of trials is fixed (n ϭ 3) It is not clear what p should be.
Do you take a national average?
Each trial is success (red) or failure (white). However p ϭ 0.5 will probably be a
If the balls are drawn with replacement good starting value for your model.
the probability of selecting a red is constant
The number of balls to be drawn is
and p ϭ _. 1
4 fixed at 3.
The trials will be independent too (if the
Selecting with replacement is critical
first is red it does not alter the probability here for R to have a binomial
of the second being red). distribution.

So R ~ B(3, _) 1
4 If the balls are selected without
replacement then p is not constant.

7

CHAPTER 1

Exercise 1B

1 The random variable X ~ B(8, _). Find
1
3
a P(X ϭ 2), b P(X ϭ 5), c P(X р 1).

2 The random variable Y ~ B(6, _). Find
1
4
a P(Y ϭ 3), b P(Y ϭ 1), c P(Y у 5).

3 The random variable T ~ B(15, _). Find
2
3
a P(T ϭ 5), b P(T ϭ 10), c P(3 р T р 4).

4 A balloon manufacturer claims that 95% of his balloons will not burst when blown up.
If you have 20 of these balloons to blow up for a birthday party
a what is the probability that none of them burst when blown up?
b Find the probability that exactly 2 balloons burst.

5 A student suggests using a binomial distribution to model the following situations.
Give a description of the random variable, state any assumptions that must be made and
give possible values for n and p.
a A sample of 20 bolts is checked for defects from a large batch. The production process
should produce 1% of defective bolts.
b Some traffic lights have three phases: stop 48% of the time, wait or get ready 4% of the
time and go 48% of the time. Assuming that you only cross a traffic light when it is in
the go position, model the number of times that you have to wait or stop on a journey
passing through 6 sets of traffic lights.
c When Stephanie plays tennis with Timothy on average one in eight of her serves is an
‘ace’. How many ‘aces’ does Stephanie serve in the next 30 serves against Timothy?

6 State which of the following can be modelled with a binomial distribution and which can
not. Give reasons for your answers.
a Given that 15% of people have blood that is Rhesus negative (RhϪ), model the number of
pupils in a statistics class of 14 who are RhϪ.
b You are given a fair coin and told to keep tossing it until you obtain 4 heads in succession.
Model the number of tosses you need.
c A certain car manufacturer produces 12% of new cars in the colour red, 8% in blue, 15%
in white and the rest in other colours. You make a note of the colour of the first 15 new
cars of this make. Model the number of red cars you observe.

7 A fair die is rolled repeatedly. Find the probability that
a the first 6 occurs on the fourth roll,
b there are 3 sixes in the first 10 rolls.

8 A coin is biased so that the probability of it landing on heads is _. The coin is tossed
2
3
repeatedly. Find the probability that
a the first tail will occur on the fifth toss,
b in the first 7 tosses there will be exactly 2 tails.
8


1.4 You can use the tables of the cumulative distribution function of the binomial
distribution to find probabilities.

In S1 you met the cumulative distribution function F(x) ϭ P(X р x).
For the binomial distribution X ~ B(n, p) there are tables giving P(X р x) for various values of n
and p. Using these tables can make potentially lengthy calculations much shorter.

Example 7

The random variable X ~ B(20, 0.4). Find
a P(X р 7), b P(X Ͻ 6), c P(X у 15).
Use n ϭ 20, p ϭ 0.4 and x ϭ 7 in
the cumulative binomial distribution
a P(X р 7) ϭ 0.4159 tables.

Always quote values from the tables
b P(X Ͻ 6) ϭ P(X р 5) in full.
ϭ 0.1256
Since X is a discrete random
variable, X Ͻ 6 means X р 5.
c P(X у 15) ϭ 1 Ϫ P(X р 14)
The tables give P(X р x) so use
ϭ 1 Ϫ 0.9984 x ϭ 5.
ϭ 0.0016
P(X у 15) ϩ P(X р 14) ϭ 1 (since
the total probability ϭ 1). The tables
only give P(X р x) so you must
The tables can also be used to ﬁnd P(X ϭ x) since write P(X у 15) as 1 Ϫ P(X р 14).
P(X ϭ x) ϭ P(X р x) Ϫ P(X р (x Ϫ 1)).

Example 8

The random variable X ~ B(25, 0.25). Find
a P(X р 6), b P(X ϭ 6), c P(X Ͼ 20), d P(6 Ͻ X р 10).

Use cumulative binomial tables with
a P(X р 6) ϭ 0.5611 n ϭ 25, p ϭ 0.25 and x ϭ 6.

b P(X ϭ 6) ϭ P(X р 6) Ϫ P(X р 5) Write P(X ϭ 6) in terms of
cumulative probabilities.
ϭ 0.5611 Ϫ 0.3783
ϭ 0.1828 Use tables.

c P(X Ͼ 13) ϭ 1 Ϫ P(X р 13) The tables give P(X р x) so you
need to remember P(X Ͼ 13) ϭ
ϭ 1 Ϫ 0.9991 P(X у 14) and use the fact that
P(X у 14) ϩ P(X р 13) ϭ 1. Then
ϭ 0.0009 use tables.

9

CHAPTER 1

Write the probability as the
d P(6 Ͻ X р 10) ϭ P(X ϭ 7, 8, 9 or 10) difference of two cumulative
ϭ P(X р 10) Ϫ P(X р 6) probabilities.

ϭ 0.9703 Ϫ 0.5611
ϭ 0.4092 Use tables.

Sometimes questions are set in context and there are many different forms of words that can
be used to ask for probabilities. The correct interpretation of these phrases is critical, especially
when dealing with discrete distributions such as the Binomial and Poisson (see Chapter 2).
The table gives some examples.

Phrase Means To use tables …

… greater than 5 … XϾ5 1 Ϫ P(X р 5)

… no more than 3 … Xр3 P(X р 3)

… at least 7 … Xу7 1 Ϫ P(X р 6)

… fewer than 10 … X Ͻ 10 P(X р 9)

… at most 8 … Xр8 P(X р 8)

Example 9

A spinner is designed so that the probability it lands on red is 0.3. Jane has 12 spins. Find the
probability that Jane obtains
a no more than 2 reds, b at least 5 reds.
Jane decides to use this spinner for a class competition. She wants the probability of winning a
prize to be Ͻ0.05. Each member of the class will have 12 spins and the number of reds will be
recorded.
c Find how many reds are needed to win a prize. Define a suitable random variable.
This makes it easier to rewrite the
question in terms of probabilities
Let X ϭ the number of reds in 12 spins. and it can help you determine the
distribution.
X ~ B(12, 0.3)
a P(X р 2) ϭ 0.2528 … no more than 2 means X р 2.

Use tables with n ϭ 12, p ϭ 0.3 and
x ϭ 2.
b P(X у 5)
… at least 5 means X у 5.
ϭ 1 Ϫ P(X р 4)
ϭ 1 Ϫ 0.7237 Since tables only give P(X р x) you
need to write as 1 Ϫ P(X р 4) and
ϭ 0.2763 then use tables with n ϭ 12,
p ϭ 0.3 and x ϭ 4.

10


Form a probability statement to
c Let r ϭ the smallest number of reds needed represent the condition for winning
to win a prize. a prize.

Require: P(X у r) Ͻ 0.05
Use the tables with n ϭ 12 and
From tables: p ϭ 0.3. You want values of x that
give cumulative probabilities Ͼ 0.95
P(X р 5) ϭ 0.8822 (since 0.95 ϭ 1 Ϫ 0.05).
P(X р 6) ϭ 0.9614
P(X р 7) ϭ 0.9905

Since x ϭ 6 gives the first value Ͼ
So: P(X р 6) ϭ 0.9614 implies that 0.95 use this probability and find
P(X у 7) ϭ 1 Ϫ 0.9614 r ϭ 7.

ϭ 0.0386
Always make sure that your final
So 7 or more reds will win a prize. answer is related back to the context
of the original question.

This technique of using the tables backwards as in part c will be met again in connection with
critical regions for hypothesis tests in Chapter 7.

Exercise 1C

1 The random variable X ~ B(9, 0.2). Find
a P(X р 4), b P(X Ͻ 3), c P(X у 2), d P(X ϭ 1).

a P(X р 10), b P(X Ͼ 7), c P(X ϭ 5), d P(2 р X р 7).

a P(X Ͻ 20), b P(X Ͼ 16), c P(11 р X р 15), d P(10 Ͻ X Ͻ 17).

a P(X Ͼ 8), b P(X р 4), c P(2 р X Ͻ 10), d P(X ϭ 4).

5 Eight fair coins are tossed and the total number of heads showing is recorded.
Find the probability of
a no heads, b at least 2 heads, c more heads than tails.

11

CHAPTER 1

6 For a particular type of plant 25% have blue flowers. A garden centre sells these plants in
trays of 15 plants of mixed colours. A tray is selected at random.
Find the probability that the number of blue flowers this tray contains is
a exactly 4, b at most 3, c between 3 and 6 (inclusive).

a the largest value of k such that P(X р k) Ͻ 0.05,
b the smallest number r such that P(X Ͼ r ) Ͻ 0.10.

a the largest value of k such that P(X Ͻ k) Ͻ 0.20,
b the smallest number r such that P(X Ͼ r) Ͻ 0.01,
c P(k р X р r).

9 In a town, 30% of residents listen to the local radio.
Ten residents are chosen at random.
a State the distribution of the random variable
X ϭ the number of these 10 residents that listen to the local radio.
b Find the probability that at least half of these 10 residents listen to local radio.
c Find the smallest value of s so that P(X у s) Ͻ 0.01.

10 A factory produces a component for the motor trade and 5% of the components are
defective. A quality control officer regularly inspects a random sample of 50 components.
Find the probability that the next sample contains
a fewer than 2 defectives,
b more than 5 defectives.
The officer will stop production if the number of defectives in the sample is greater than a
certain value d. Given that the officer stops production less than 5% of the time.
c find the smallest value of d.

1.5 You can use simple formulae to find the mean and variance of the binomial
distribution.

If X ~ B(n, p) then

E(X) ϭ ␮ ϭ np
Var(X) ϭ ␴ 2 ϭ np(1 Ϫ p)

These formulae are given in the formula booklet. You are not expected to know how to prove
these formulae in the S2 exam.

12


Example 10

A fair, 4-sided die has the numbers 1, 2, 3 and 4 on its faces. The die is rolled 20 times.
The random variable X represents the number of 4s obtained.

a Find the mean and variance of X.

b Find P(X , ␮ Ϫ ␴).

Identify the distribution of X. There
X ~ B(20, 0.25) are 20 rolls of the die so n ϭ 20 and
a binomial is appropriate.

a E(X) ϭ ␮ ϭ 20 ϫ 0.25 ϭ 5 Use the formulae with n ϭ 20 and
Var (X) ϭ ␴ 2 ϭ 20 ϫ 0.25 ϫ 0.75 ϭ __ or 3.75
15 p ϭ 0.25.
4

____
0.75 ϭ 1 Ϫ 0.25 ϭ 1 Ϫ p
b ␴ ϭ √3.75 ϭ 1.936 …
_____
P(X Ͻ ␮ Ϫ ␴) ϭ P(X Ͻ 5 Ϫ 1.936 …) Remember Var(X) ϭ ␴2 ∴ ␴ ϭ √3.75 .

ϭ P(X Ͻ 3.06 …) Since X is discrete, X Ͻ 3.06 …
ϭ P(X р 3) means X р 3.

ϭ 0.2252 Use tables with n ϭ 20, p ϭ 0.25
and x ϭ 3.

Example 11

David believes that 35% of people in a certain town will vote for him in the next election and he
commissions a survey to verify this. Find the minimum number of people the survey should ask
to have a mean number of more than 100 voting for David.

Define a suitable random variable.
Let n ϭ the number of people asked. The proportion of 35% will give
Let X ϭ the number of people (out of n) voting for p ϭ 0.35.

David.
X ~ B(n, 0.35) Use the formula ␮ ϭ np.
E(X ) ϭ 0.35n
So 0.35n Ͼ 100 Form a suitable inequality, using the
100 fact that the mean is greater than
n Ͼ _____ ϭ 285.7 … 100, and solve to find n.
0.35
So (n ϭ) 286 people should be asked.

13

CHAPTER 1

Example 12

An examiner is trying to design a multiple choice test. For students answering the test at random,
he requires that the mean score on the test should be 20 and the standard deviation should be
at least 4. Assuming that each question has the same number of alternative answers, ﬁnd how
many questions and how many alternative answers each question should have. The number of
alternatives for each question should be as few as possible.

Let n ϭ the number of questions Define n and p to form a
suitable binomial distribution.
and p ϭ the probability of guessing a correct answer.
Let X ϭ the number of questions answered correctly
at random
X ~ B(n, p)
Use the formulae for mean
E(X ) ϭ 20 so np ϭ 20 and variance. Remember
variance ϭ (st. dev)2.
St. dev у 4 so np(1 Ϫ p) у 42
So 20(1 Ϫ p) у 16
1 Ϫ p у 0.8 Substitute np ϭ 20 and solve
for p.
0.2 у p
So the examiner should use p ϭ 0.2, i.e. have 5
alternatives for each question.
np ϭ 20 with p ϭ 0.2 gives n ϭ 100.
So the test should have 100 questions and 5
alternative answers for each.

Exercise 1D

1 A fair cubical die is rolled 36 times and the random variable X represents the number of
sixes obtained. Find the mean and variance of X.
2 a Find the mean and variance of the random variable X ~ B(12, 0.25).
b Find P(␮ Ϫ ␴ Ͻ X Ͻ ␮ ϩ ␴).
3 a Find the mean and variance of the random variable X ~ B(30, 0.40).
b Find P(␮ Ϫ ␴ Ͻ X р ␮).
4 It is estimated that 1 in 20 people are left-handed.
a What size sample should be taken to ensure that the expected number of left-handed
people in the sample is 3?
b What is the standard deviation of the number of left-handed people in this case?
5 An experiment is conducted with a fair die to examine the number of sixes that occur. It
is required to have the standard deviation smaller than 1. What is the largest number of
throws that can be made?
6 The random variable X ~ B(n, p) has a mean of 45 and standard deviation of 6.
Find the value of n and the value of p.

14


1.6 You can solve harder problems.

Example 13

In Joe’s café 70% of customers buy a cup of tea.
a In a random sample of 20 customers find the probability that more than 15 buy a cup of tea.
The proportion of customers who buy a chocolate muffin at Joe’s café is 0.35.
b Find the probability that Joe sells the first chocolate muffin to his fifth customer.
Assuming that buying a chocolate muffin and buying a cup of tea are independent events
c find the probability that in a random sample of 20 customers more than 15 buy a cup of tea
and at least 6 buy a chocolate muffin.

a Let X ϭ the number out of 20 who buy a cup Since 0.7 is not in the tables
you will need to consider the
of tea.
complementary random variable Y.
X ~ B(20, 0.7)
Let Y ϭ the number of customers who do Note that X Ͼ 15 is the same as
Y р 4 and then use tables with
not buy a cup of tea. n ϭ 20 and p ϭ 0.3.
Y ~ B(20, 0.3)
P(X Ͼ 15) ϭ P(Y р 4) ϭ 0.2375 Define the random variable carefully.
Note the variable is the number of
b Let N ϭ the number of the first customer the customer not the number of
muffins.
who buys a chocolate muffin.
If C represents the event that a person buys
N is not binomial since there is no
a chocolate muffin then the first 5 fixed value for n.
customers must be CЈCЈCЈCЈC
So P(N ϭ 5) ϭ (0.65)4 ϫ 0.35 There is only one case possible so
ϭ 0.062477 … no ( n ) term.
r
ϭ 0.0625 (3 s.f.)
c From part a P(X Ͼ 15) ϭ 0.2375
The probability for cups of tea was
Let M ϭ number of customers out of 20 who
found in a. So define the variable M.
buy a chocolate muffin
M ~ B(20, 0.35)
Use P(M у 6) ϩ P(M р 5) ϭ 1 and
P(M у 6) ϭ 1 Ϫ P(M р 5) then use tables with n ϭ 20 and
p ϭ 0.35.
ϭ 1 Ϫ 0.2454
ϭ 0.7546
To find the probability of both
So P(M у 6 and X Ͼ 15) ϭ 0.7546 ϫ 0.2375 X Ͼ 15 and M у 6 you multiply the
probabilities together since buying
ϭ 0.17921 …
a chocolate muffin and a cup of tea
ϭ 0.179 (3 s.f.) are independent.

15

CHAPTER 1

Example 14

Yummy sweets are sold in tubes of 15 sweets per tube. The sweets are of different flavours and
20% are blackcurrant.
a Show that the probability of a randomly chosen tube of Yummy sweets containing 3 or more
blackcurrant ones is 0.602.
Robert buys 5 tubes of Yummy sweets.
b Find the probability that at least 4 of the tubes contain 3 or more blackcurrant sweets.

a Let X ϭ number of blackcurrant sweets in a Define the random variable carefully.
tube of 15
X ~ B(15, 0.20)
P(X у 3) ϭ 1 Ϫ P(X р 2) Use P(X у 3) ϩ P(X р 2) ϭ 1 and
tables with n ϭ 15 and p ϭ 0.2.
ϭ 1 Ϫ 0.3980
ϭ 0.602
Define a new random variable.
b Y ϭ number of packets containing 3 or more Note that you are now counting
blackcurrant sweets packets not blackcurrant sweets. This
changes n and p.
Y ~ B(5, 0.602)
P(Y у 4) ϭ P(Y ϭ 4) ϩ P(Y ϭ 5)

ϭ 5 (0.602)4 ϫ 0.398 ϩ (0.602)5 Since p ϭ 0.602 is not in the tables
( )
4 you will have to use the formula.
ϭ 0.34042 …
ϭ 0.340 (3 s.f.)

Mixed exercise 1E

1 A coin is biased so that the probability of a head is _. The coin is tossed repeatedly.
2
3
Find the probability that
a the first tail will occur on the sixth toss,
b in the first 8 tosses there will be exactly 2 tails.

2 Records kept in a hospital show that 3 out of every 10 patients who visit the accident and
emergency department have to wait more than half an hour. Find, to 3 decimal places, the
probability that of the first 12 patients who come to the accident and emergency department
a none, b more than 2,
will have to wait more than half an hour.

3 A factory is considering two methods of checking the quality of production of the batches of
items it produces.
Method I A random sample of 10 items is taken from a large batch and the batch is accepted
if there are no defectives in this sample. If there are 2 or more defectives the batch is
16


rejected. If there is only 1 defective then another sample of 10 is taken and the batch is
accepted if there are no defectives in this second sample, otherwise the whole batch is
rejected.
Method II A random sample of 20 items is taken from a large batch and the batch is accepted
if there is at most 1 defective in this sample, otherwise the whole batch is rejected.
The factory knows that 1% of items produced are defective and wishes to use the method of
checking the quality of production for which the probability of accepting the whole batch is
largest.
a Decide which method the factory should use.
b Determine the expected number of items sampled using Method I.

4 a State clearly the conditions under which it is appropriate to assume that a random
variable has a binomial distribution.
A door-to-door canvasser tries to persuade people to have a certain type of double glazing
installed. The probability that his canvassing at a house is successful is 0.05.
b Find the probability that he will have at least 2 successes out of the first 10 houses he
canvasses.
c Find the number of houses he should canvass per day in order to average 3 successes per
day.
d Calculate the least number of houses that he must canvass in order that the
probability of his getting at least one success exceeds 0.99. E

5 An archer fires arrows at a target and for each arrow, independently of all the others, the
probability that it hits the bull’s eye is _.
1
8
a Given that the archer fires 5 arrows, find the probability that fewer than 2 arrows hit the
bull’s-eye.
The archer fires 5 arrows, collects them from a target and fires all 5 again.
b Find the probability that on both occasions fewer than 2 hit the bull’s eye.

6 A completely unprepared student is given a true/false type test with 10 questions.
Assuming that the student answers all the questions at random
a find the probability that the student gets all the answers correct.
It is decided that a pass will be awarded for 8 or more correct answers.
b Find the probability that the student passes the test.

7 A six-sided die is biased. When the die is thrown the number 5 is twice as likely to appear
as any other number. All the other faces are equally likely to appear. The die is thrown
repeatedly. Find the probability that
a the first 5 will occur on the sixth throw,
b in the first eight throws there will be exactly three 5s. E

8 A manufacturer produces large quantities of plastic chairs. It is known from previous records
that 15% of these chairs are green. A random sample of 10 chairs is taken.
17

CHAPTER 1

a Define a suitable distribution to model the number of green chairs in this sample.
b Find the probability of at least 5 green chairs in this sample.
c Find the probability of exactly 2 green chairs in this sample.

9 A bag contains a large number of beads of which 45% are yellow. A random sample of 20
beads is taken from the bag. Use the binomial distribution to find the probability that the
sample contains
a fewer than 12 yellow beads,
b exactly 12 yellow beads. E

Summary of key points
1 If X ~ B(n, p) then

P(X ϭ x) ϭ ( n ) p x (1 Ϫ p)n Ϫ x
x

2 If X ~ B(n, p) then
E(X) ϭ np
Var(X) ϭ np (1 Ϫ p)

3 Conditions for a binomial distribution
• A fixed number of trials, n.
• Each trial should be success or failure.
• The trials are independent.
• The probability of success, p, at each trial is constant.

18

Chapter1 draft

More Related Content

Similar to Chapter1 draft (20)

Recently uploaded (20)

Chapter1 draft