Probability theory good

What do you mean by probably ?
meaning not sure ! Or are un-certain
e.g. probably it will rain today or probably
Pakistan will win the match. In such and
many other situations, there is an element of
un-certainty in our statements.
In Statistics, we have a technique or tool which is
used to measure the amount of un-certainty. i.e.
PROBABILITY.

The range of this numerical measure is from zero to one.
i.e. if Ei is any event, then
0 ≤ P(Ei) ≤ 1 for each i
The word Probability has two basic meanings:
1: It is the quantitative measure of un-certainty
2: Its the measure of degree of belief in a
particular statement.
Example of a melon picked from a carton.

The first meaning is related to the problems
of objective approach.
The second is related to the problems of
subjective approach.

There is a situation, where probability is an
inherent part e.g. we have:
Population Sample
Here we have full Here we have
Information and have partial information
certain results. and have un-certain
results.

As we know, mostly the problems are solved
using the sample data.
A statistical technique which deals with the
sample Results is statistical inference.
Un-certainty (due to sample data) is also an
inherent part of statistical inference.

Statistics and Probability theory
constitutes a branch of mathematics for
dealing with uncertainty
Probability theory provides a basis for the
science of statistical inference from
data

…a random experiment is an action or
process that leads to one of several possible
outcomes. For example:
6.
8
Experiment Outcomes
Flip a coin Heads, Tails
Exam Marks Numbers: 0, 1, 2, ..., 100
Roll a die 1,2,3,4,5,6
Course Grades F, D, C, B, A, A+

List the outcomes of a random experiment…
List: “Called the Sample Space”
Outcomes: “Called the Simple Events”
This list must be exhaustive, i.e. ALL
possible outcomes included.
Die roll {1,2,3,4,5} Die roll {1,2,3,4,5,6}
6.9

The list must be mutually exclusive,
i.e. no two outcomes can occur at the
same time:
Die roll {odd number or even number}
Die roll{ number less than 4 or even
number}

A list of exhaustive
[don’t leave anything out]
and
mutually exclusive outcomes
[impossible for 2 different events to occur in
the same experiment]
is called a sample space and is denoted by
S.

A usual six-sided die has a sample space
S={1,2,3,4,5,6}
If two dice are rolled ( or, equivalently,
if one die is rolled twice)
The sample space is shown in Figure 1.2.

An individual outcome of a sample space is
called a simple event
[cannot break it down into several other
events],
An event is a collection or set of one or
more simple events in a sample space.
6.14

Roll of a die: S = {1, 2, 3, 4, 5, 6}
Simple event: the number “3” will be rolled
Compound Event: an even number
(one of 2, 4, or 6) will be rolled

The probability of an event is the sum of the
probabilities of the simple events that constitute
the event.
E.g. (assuming a fair die) S = {1, 2, 3, 4, 5, 6}
and
P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6
Then:
P(EVEN) = P(2) + P(4) + P(6) = 1/6 + 1/6 +1/6
= 3/6 = 1/2
6.16

Types of an Event
Impossible
Event
Possible
Event
Simple Event Compound Event
Certain Event
Mutually
Exclusive Equally
Likely
Exhaustive Dependent Independent

A desired subset of a sample space
containing at least one possible outcome.
Example
When a die is rolled once, Sample Space is
S={ 1,2,3,4,5,6}
Event E is defined as an odd number
appears on upper side of a die
E={1,3,5}

containing only one possible outcome.
Example
Event D is defined as head will turn up on a
coin
D={ H}

containing at least two possible outcome.
Example
Event E is defined as an odd number
appears on upper side of a die
E={1,3,5}

A desired sub set of sample space
containing all possible outcomes. This
event is also called certain event.
Example
Any number from 1 to 6 appears on upper
side of a die.
S={1,2,3,4,5,6}

Mutually Exclusive Events
Two events A and B defined in a sample
space S and have nothing common between
them are called mutually exclusive events.
Example
The events A (number 4 appear on a die)
and B( an odd number appear on a die) are
two mutually exclusive events.

Equally Likely Events
Two events A and B defined in simple
sample space S and their chances of
occurrences are equal are called equally
likely events.
Example
The events A (head will turn up on a coin)
and B (tail will turn up on a coin).

Two events A and B defined in simple sample
space S
(i) Nothing common between them.
(ii) Their union is same as the sample space.
Example
The events A(head will turn up on a coin) and B
( tail will turn up on a coin)
are two exhaustive events as A intersection B is
empty set as well as their union is S.

Two events A and B are independent events
if occurrence or non occurrence of one does
not affect the occurrence or non-occurrence
of the other.

Example: The event
A (get king card in first attempt)
and
B(get queen card in second attempt )
when two cards are to be drawn in
succession replacing the first drawn card in
pack before second draw.

Two events A and B are dependent events if
occurrence or non occurrence of one affect
the occurrence or non-occurrence of the
other.

Example : The event
A (get king card in first attempt)
and
B(get queen card in second attempt )
when two cards are to be drawn in
succession not replacing the first drawn
card in pack before second draw

There are three ways to assign a probability,
P(Ei), to an outcome, Ei, namely:
Classical approach:
make certain assumptions (such as equally
likely, independence) about situation.
Relative frequency:
assigning probabilities based on
experimentation or historical data.
Subjective approach:
Assigning probabilities based on the assignor’s
judgment.
6.30

If an experiment has “n” possible outcomes
[all equally likely to occur], this method
would assign a probability of 1/n to each
outcome.
Experiment: Rolling a die
Sample Space: S = {1, 2, 3, 4, 5, 6}
Probabilities: Each sample point has a 1/6
chance of occurring.
6.31

If an experiment results in ‘n’ mutually
exclusive, equally likely and exhaustive
events and if ‘m’ of these are in favor of the
occurrence of an event ‘A’, then probability
of event A is written as:
P(A)=m / n

Experiment: Rolling 2 die [dice] and
summing 2 numbers on top.
Sample Space: S = {2, 3, …, 12}
Probability Examples:
P(2) = 1/36
P(6) = 5/36
P(10) = 3/36
6.33
1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12

Relative frequency of occurrence is based
on actual observations, it defines probability
as the number of times, the relevant event
occurs, divided by the number of times the
experiment is performed in a large number
of trials.

Bits & Bytes Computer Shop tracks the number
of desktop computer systems it sells over a
month (30 days):
For example,
10 days out of 30
2 desktops were sold.
From this we can construct
the 111111111probabilities of an event
(i.e. the # of desktop sold on a given day)…
6.35
Desktops Sold # of Days
0 1
1 2
2 10
3 12
4 5

“There is a 40% chance Bits & Bytes will sell 3 desktops on any
given day” [Based on estimates obtained from sample of 30
days]
6.36
Deskto
ps Sold
[X]
# of
Days
Relative
frequency
Desktops Sold
0 1 1/30 = .03 .03 =P(X=0)
1 2 2/30 = .07 .07 = P(X=1)
2 10 10/30 = .33 .33 = P(X=2)
3 12 12/30 = .40 .40 = P(X=3)
4 5 5/30 = .17 .17 = P(X=4)
30 ∑ = 1.00 ∑ = 1.00

“In the subjective approach we define
probability as the degree of belief that we
hold in the occurrence of an event”
P(you drop this course)
P(NASA successfully land a man on the
moon)
P(girlfriend says yes when you ask her to
marry you)
6.37

We study methods to determine probabilities
of events that result from combining other
events in various ways.
6.38

There are several types of combinations and
relationships between events:
Complement of an event [everything other
than that event]
Intersection of two events [event A and event B]
or [A*B]
Union of two events [event A or event B]
or [A+B]
6.39

Why are some mutual fund managers more
successful than others? One possible factor
is where the manager earned his or her
MBA.
The following table compares mutual fund
performance against the ranking of the
school where the fund manager earned their
MBA: Where do we get these probabilities
from? [population or sample?]
6.40

Venn Diagrams
6.41
Mutual fund outperforms
the market
Mutual fund doesn’t
outperform the market
Top 20 MBA program .11 .29
Not top 20 MBA program .06 .54
E.g. This is the probability that a mutual fund
outperforms AND the manager was in a top-20 MBA
program; it’s a joint probability [intersection].

Alternatively, we could introduce shorthand
notation to represent the events:
A1 = Fund manager graduated from a top-20 MBA program
A2 = Fund manager did not graduate from a top-20 MBA program
B1 = Fund outperforms the market
B2 = Fund does not outperform the market
6.42
B1 B2
A1
.11 .29
A2
.06 .54
E.g. P(A2 and B1) = .06
= the probability a fund outperforms the market
and the manager isn’t from a top-20 school.

Marginal probabilities are computed by
adding across rows and down columns; that is
they are calculated in the margins of the table:
6.43
B1 B2 P(Ai)
A1
.11 .29 .40
A2
.06 .54 .60
P(Bj) .17 .83 1.00
P(B1) = .11 + .06
P(A2) = .06 + .54
“what’s the probability a fund
outperforms the market?”
“what’s the probability a fund
manager isn’t from a top school?”
BOTH margins must add to 1
(useful error check)

Conditional probability is used to determine
how two events are related; that is, we can
determine the probability of one event given the
occurrence of another related event.
6.44

Experiment: random select one student in
class.
P(randomly selected student is male/student
is on 3rd row) =
Conditional probabilities are written as
P(A/ B) and read as “the probability of
A given B”
6.45

Again, the probability of an event given that
another event has occurred is called a
conditional probability…
P( A and B) = P(A)*P(B/A) = P(B)*P(A/B)
both are true
6.46

Events:
A1 = Fund manager graduated from a top-20 MBA
program
A2 = Fund manager did not graduate from a top-20
MBA program
6.47
B1 B2
A1
.11 .29
A2
.06 .54

Example 2 • What’s the probability that a fund will
outperform the market given that the manager
graduated from a top-20 MBA program?
Recall:
A1 = Fund manager graduated from a top-20 MBA program
A2 = Fund manager did not graduate from a top-20 MBA
program
Thus, we want to know “what is P(B1 | A1) ?”
6.48

We want to calculate P(B1 | A1)
6.49
Thus, there is a 27.5% chance that that a fund will outperform the
market given that the manager graduated from a top-20 MBA
program.
B1 B2 P(Ai)
A1
.11 .29 .40
A2
.06 .54 .60
P(Bj) .17 .83 1.00

One of the objectives of calculating conditional
probability is to determine whether two events are
related.
In particular, we would like to know whether they are
independent, that is, if the probability of one event
is not affected by the occurrence of the other event.
Two events A and B are said to be independent if
P(A|B) = P(A)
and
P(B|A) = P(B)
6.50

For example, we saw that
P(B1 | A1) = .275
The marginal probability for B1 is: P(B1) = 0.17
Since P(B1|A1) ≠ P(B1), B1 and A1 are not
independent events.
Stated another way, they are dependent. That
is, the probability of one event (B1) is affected
by the occurrence of the other event (A1).
6.51

We introduce three rules that enable us to calculate
the probability of more complex events from the
probability of simpler events…
The Complement Rule – May be easier to calculate
the probability of the complement of an event and
then substract it from 1.0 to get the probability of the
event.
P(at least one head when you flip coin 100 times)
= 1 – P(0 heads when you flip coin 100 times)
The Multiplication Rule: P(A*B)
The Addition Rule: P(A+B)
6.52

Multiplication Rule for Independent Events: Let A and B be two
independent events, then
( ) ( ) ( ).P A B P A P B 
Examples:
• Flip a coin twice. What is the probability of observing two heads?
• Flip a coin twice. What is the probability of getting a head and then a
tail? A tail and then a head? One head?
• Three computers are ordered. If the probability of getting a “working”
computer is .7, what is the probability that all three are “working” ?

A graduate statistics course has seven male and
three female students. The professor wants to select
two students at random to help her conduct a
research project. What is the probability that the two
students chosen are female?
P(F1 * F2) = ???
Let F1 represent the event that the first student is
female
P(F1) = 3/10 = .30
What about the second student?
P(F2 /F1) = 2/9 = .22
P(F1 * F2) = P(F1) * P(F2 /F1) = (.30)*(.22) = 0.066
NOTE: 2 events are NOT independent.
6.54

The professor in Example is unavailable. Her
replacement will teach two classes. His style is
to select one student at random and pick on
him or her in the class. What is the probability
that the two students chosen are female?
Both classes have 3 female and 7 male
students.
P(F1 * F2) = P(F1) * P(F2 /F1) = P(F1) * P(F2)
= (3/10) * (3/10) = 9/100 = 0.09
NOTE: 2 events ARE independent.
6.55

Addition rule provides a way to compute the
probability of event A or B or both A and B
occurring; i.e. the union of A and B.
P(A or B) = P(A + B) = P(A) + P(B) –
P(A and B)
Why do we subtract the joint probability P(A
and B) from the sum of the probabilities of A
and B?
6.56
P(A or B) = P(A) + P(B) – P(A and B)

P(A1) = .11 + .29 = .40
P(B1) = .11 + .06 = .17
By adding P(A) plus P(B) we add P(A and B) twice. To
correct we subtract P(A and B) from P(A) + P(B)
6.57
B1 B2 P(Ai)
A1
.11 .29 .40
A2
.06 .54 .60
P(Bj) .17 .83 1.00
P(A1 or B1) = P(A) + P(B) –P(A and B) = .40 + .17 - .11 = .46
B1
A1

If and A and B are mutually exclusive the
occurrence of one event makes the other one
impossible. This means that
P(A and B) = P(A * B) = 0
The addition rule for mutually exclusive events
is
P(A or B) = P(A) + P(B)
Only if A and B are Mutually Exclusive.
6.58

In a large city, two newspapers are published, the Sun
and the Post. The circulation departments report that
22% of the city’s households have a subscription to the
Sun and 35% subscribe to the Post. A survey reveals
that 6% of all households subscribe to both newspapers.
What proportion of the city’s households subscribe to
either newspaper?
That is, what is the probability of selecting a household at
random that subscribes to the Sun or the Post or both?
P(Sun or Post) = P(Sun) + P(Post) – P(Sun and Post)
= .22 + .35 – .06 = .51
6.59

Probability theory good

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Probability theory good