Introduction to Probability and Random Variables and Discrete probability Distributions

Chapter 5
Introduction to
Probability

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6.1 Assigning probabilities to Events
A random experiment is a process or course
of action, whose outcome is uncertain.
Experiment
• Flip a coin
• The marks of a statistics test
• The time to assemble
a computer

Outcomes
Heads and Tails
Numbers between 0 and 10
Non-negative numbers

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• Repeated random experiments may result in different
outcomes. One can only refer to the experiment outcome
in terms of the probability of each outcome to occur.

• To determine the probabilities we need to define the
sample space,
– which is exhaustive (list of all the possible

outcomes).
– in which the outcomes are mutually exclusive
(outcome do not overlap).
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Example 1: Build the sample space for the two random
experiments described below

Case 1: A ball is randomly selected from a bag
containing blue and red balls. An outcome is
considered the ball color.
The sample space: {B, R}

Case 2: Two balls are randomly selected from a
bag that contains blue and red balls. An outcome is
considered the colors of the two balls drawn.
The sample space: {BB, BR, RB, RR}
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Example 2: The random experiment: randomly
select two numbers from the set 1, 2, 3, 4, 5 (a
number cannot be selected twice). An outcome
is defined by the sum of the two numbers.

1+3

1+5, 2+4

3+5

The sample space: {3, 4, 5, 6, 7, 8, 9}
1+2 1+4, 2+3;

2+5, 3+4

4+5
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Sample Space: S = {O1, O2,
…,Ok}
O1

O2

Simple events: The individual outcomes that cannot
be further decomposed
An event is a collection of simple events.
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Example 3: A dice is rolled once. Specify the
sample points that belong to each event.
Event A = the number facing up is 6
Event B = The number facing up is odd
Event C = The number facing up is not
greater than 4
Solution: A = {6};B = {1, 3, 5}; C = {1, 2, 3)

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Example 4: A dice is rolled once.
Event A = the number facing up is 6
Event B = The number facing up is odd
Event C = The number facing up is less than 4
The outcome is 3. Which event takes place?
Solution: Event B and event C take place
because the outcome ‘3’ belongs to both
events.

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Sample Space: S = {O1, O2,
…,Ok}
Our objective
objective isis to
to
Our
determine P(A),
P(A), the
the
determine
probability that
that event
event AA
probability

will occur.
occur.
will

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6.1 Assigning probabilities to
Events

Given a sample space S={O1,O2,…,Ok},
The probability of a simple event P(Oi): the following
characteristics must hold:

1.
2.

0 P(O ) 1 for each i
i
n
 P(Oi ) 1
i 1

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6.2 Joint, Marginal, and
Conditional Probability
The probability of an event: The probability
P(A) of event A is the sum of the probabilities

assigned to the simple events in A.
Understanding relationships among events may
reduce the computational effort in determining
the probability of events.
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The intersection
The intersection of event A and B, denoted by “A
and AB”(B
), is the event that both A and B occur.

C
A

B

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Two events are said to be mutually exclusive if the
occurrence of one precludes the occurrence of the
other one.
If A and B are mutually exclusive, by definition, the
probability of their intersection is equal to zero.
Example: When rolling a dice once the event “The
number facing up is 6” and the event “The number
facing up is odd” are mutually exclusive.

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The Joint Probability
The Probabilities of Joint Events
The probability of the intersection of A and B is
called also the joint probability of A and B = P(A
and B).

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Example 9: A potential investor examined the
relationship between the performance of
mutual funds and the school the fund manager
earned his/her MBA. The following table
describes the joint probabilities.
Mutual fund
outperforms the
market
(B1)

Mutual fund
doesn’t outperform
the market
(B2)

Top 20 MBA program
(A1)

.11

.29

Not top 20 MBA program
(A2)

.06

.54
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Marginal Probabilities
Observe first the following demonstration
Let us separate event C
intersect
into two subEvent
event:C “A
and with
both
C”, and “B
andevent
C” A and B

A

C

B

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A

A and C

B and C

B

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The probability of event C can be calculated as
the sum of the two joint probabilities.

A

C

B

P(C) = P(A and C) + P(B and C))
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Applying this concept to the table of joint
probabilities, the marginal probabilities are
determining by adding joint probabilities across

rows and columns.
Mutual fund
Mutual fund
Marginal
outperforms the doesn’t outperform Prob.
market (B1)
the market (B2)
P(Ai)
Top 20 MBA program (A1)

P(A1 and B1) +

P(A1 and B2)

= P(A1)

Not top 20 MBA program (A2)

P(A2 and B1) +

P(A2 and B2)

= P(A2)

Marginal Probability P(Bj)
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Mutual fund
Mutual fund

Marginal
outperforms the doesn’t outperform Prob.
market (B1)
the market (B2)
P(Ai)
Top 20 MBA program (A1)

.11

+

.29

=

.40

Not top 20 MBA program (A2)

.06

+

.54

=

.60

Marginal Probability P(Bj)

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Mutual fund
Mutual fund
Marginal
outperforms the doesn’t outperform Prob.
market (B1)
the market (B2)
P(Ai)

P(A1 and B1)
+
Not top 20 MBA program (A2) P(A and B
2
1
=
Marginal Probability P(Bj)
P(B1)
Top 20 MBA program (A1)

P(A1 and B2)
+
P(A2 and B2
=
P(B2)

.40
.60

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Mutual fund
Mutual fund
Marginal
outperforms the doesn’t outperform Prob.
market (B1)
the market (B2)
P(Ai)
Top 20 MBA program (A1)
Not top 20 MBA program (A2)

.11
+

.06

.29
+

Marginal Probability P(Bj)

.17

.83

.54

.40
.60

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Conditional Probability
Frequently, information about the occurrence of
event A changes the probability of event B.

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The sample space is
reduced from ‘S’ to ‘A’.

The sample space is S.

P(B) is
{The area of B}
{The area of S}

S
This is event B

B

A

The probability of

event given ‘A’ is
roughly:

Now, assume event ‘A’
takes place before event
B.
Note that event B is
contained in event A.
P(B) is not equal to P(B given A)

{The area of B}
{The area of A}.


Specifically, the conditional probability of event
B given that event A has occurred is calculated
as follows:
P(A and B)
P(B|A) =
P(A)

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