Basic Concepts of Probability and Counting
Section 3.2
Summer 2013 - Math 1040
June 19
(1040)
M 1040 - 3.2
June 19
1 / 10
Roadmap
Conditional Probability and the Multiplication Rule
Pages 145 - 149
I
Conditional probability that an event occurs when another event
occurs.
I
Independent and dependent events.
I
The multiplication rule for joint events.
(1040)
M 1040 - 3.2
June 19
2 / 10
Conditional Probability
The conditional probability of an event is the probability that it will
occur, given that another event has already occured. It is denoted
P(B|A)
(1040)
”the probability of B, given A.”
M 1040 - 3.2
June 19
3 / 10
Conditional Probability
Example 1, p 145 The table below shows the results of a study on
children’s IQ and the presence of a specific gene in the child.
High IQ
Low IQ
Total
Gene Present
33
39
72
Gene Not Present
19
11
30
Total
52
50
102
Find the probability that child has a high IQ, given that the child has the
gene, and then find the probability that child has a high IQ, given that the
child does not have the gene.
(1040)
M 1040 - 3.2
June 19
4 / 10
Independence and dependence
Two events are independent if the occurence of the events do not affect
one another. If this is so, then
P(B|A) = P(B) and P(A|B) = P(A).
Events that are not independent are dependent. That is
P(B|A) 6= P(B) and P(A|B) 6= P(A)
(1040)
M 1040 - 3.2
June 19
5 / 10
Independence and dependence
Example 2, p 146 Determine whether the events are independent or
dependent.
1. Selecting a king from a deck of cards (A), not replacing it, and then
selecting a queen from the deck (B).
2. Tossing a coin and getting a tails (A), and then rolling a six-sided die
and obtaining a six (B).
3. Driving over 85 miles per hour (A), and then getting in a car accident
(B).
1. dependent, 2. independent, 3. dependent.
(1040)
M 1040 - 3.2
June 19
6 / 10
The Multiplication Rule (General)
The probability that two events A and B will occur in sequence is
P(A and B) = P(A) · P(B|A)
1. Find the probability that the first event A occurs. (Find P(A).)
2. Find the conditional probability the second event B occurs given that
the first event A occurs. (Find P(B|A).)
3. Multiply the two probabilities.
(1040)
M 1040 - 3.2
June 19
7 / 10
The Multiplication Rule (General)
The probability that two independent events A and B will occur in
sequence is
P(A and B) = P(A) · P(B)
1. Events A and B must be independent.
2. Find the probability that the first event A occurs. (Find P(A).)
3. Find the probability the second event B occurs. (Find P(B).)
4. Multiply the two probabilities.
(1040)
M 1040 - 3.2
June 19
8 / 10
Multiplication Rule
Example - Try It Yourself 3, page 147
1. The probability that a salmon swims successfully through a dam is
0.85. Find the probability that two salmon swim successfully through
a dam.
2. Two cards are selected from a standard deck of cards without
replacement. Find the probability that they are both hearts.
1. independent (0.85)2 = 0.7225
2. dependent
13 12
·
= 0.0588.
52 51
(1040)
M 1040 - 3.2
June 19
9 / 10
Assignments
Assignment:
1. Read pages 145-149
2. Page 150, 1 -33 odd
3. Exercise handout for 3.2.
Vocabulary: conditional probabilities, independent and dependent events,
multiplication rule
Understand: Find conditional probabilities, tell between independent and
dependent events. Use the multiplication rule for joint events, and the two
versions.
(1040)
M 1040 - 3.2
June 19
10 / 10