Download A and B

Elementary
Probability Theory
5
Copyright © Cengage Learning. All rights reserved.
Section
5.2
Some Probability
Rules—Compound
Events
Copyright © Cengage Learning. All rights reserved.
Focus Points
•
Compute probabilities of general compound
events.
•
Compute probabilities involving independent
events or mutually exclusive events.
•
Use survey results to compute conditional
probabilities.
3
Conditional Probability and Multiplication Rules
4
Conditional Probability and Multiplication Rules
Independent events
5
Conditional Probability and Multiplication Rules
Conditional probability P(A|B)
If the events are dependent, then we must take into
account the changes in the probability of one event caused
by the occurrence of the other event.
The notation P(A, given B) denotes the probability that
event A will occur given that event B has occurred.
This is called a conditional probability.
6
Conditional Probability and Multiplication Rules
We read P(A, given B) as “probability of A given B.” If A
and B are dependent events, then P(A)  P(A, given B)
because the occurrence of event B has changed the
probability that event A will occur.
A standard notation for P(A, given B) is P(A | B).
7
Conditional Probability and Multiplication Rules
Multiplication rules of probability
We will use either formula (5) or formula (6) according to
the information available.
Formulas (4), (5), and (6) constitute the multiplication rules
of probability.
They help us compute the probability of events happening
together when the sample space is too large for convenient
reference or when it is not completely known.
8
Conditional Probability and Multiplication Rules
Note: For conditional probability, observe that the
multiplication rule
P(A and B) = P(B)  P(A | B)
can be solved for P(A | B), leading to
9
Example 4 – Multiplication Rule, Independent Events
Suppose you are going to throw two fair dice. What is the
probability of getting a 5 on each die?
Solution Using the Multiplication Rule:
The two events are independent, so we should use
formula (4).
P(5 on 1st die and 5 on 2nd die)
= P(5 on 1st)  P(5 on 2nd)
To finish the problem, we need to compute the probability
of getting a 5 when we throw one die.
10
Example 4 – Solution
cont’d
There are six faces on a die, and on a fair die each is
equally likely to come up when you throw the die. Only one
face has five dots, so by formula (2) for equally likely
outcomes,
Now we can complete the calculation.
P(5 on 1st die and 5 on 2nd die) = P(5 on 1st)  P(5 on 2nd)
11
Example 4 – Solution
cont’d
Solution Using Sample Space:
The first task is to write down the sample space. Each die
has six equally likely outcomes, and each outcome of the
second die can be paired with each of the first. The sample
space is shown in Figure 5-2.
Sample Space for Two Dice
Figure 5-2
12
Example 4 – Solution
cont’d
The total number of outcomes is 36, and only one is
favorable to a 5 on the first die and a 5 on the second. The
36 outcomes are equally likely, so by formula (2) for equally
likely outcomes,
P(5 on 1st and 5 on 2nd) =
The two methods yield the same result. The multiplication
rule was easier to use because we did not need to look at
all 36 outcomes in the sample space for tossing two dice.
13
Conditional Probability and Multiplication Rules
Procedure:
14
15
16
17
18