Chapter 5
Section 4
Conditional Probability and the
General Multiplication Rule
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 4 – Slide 1 of 14
Chapter 5 – Section 4
● Learning objectives
1 Compute conditional probabilities
2 Compute probabilities using the General Multiplication
Rule
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 4 – Slide 2 of 14
Chapter 5 – Section 4
● Learning objectives
1 Compute conditional probabilities
2 Compute probabilities using the General Multiplication
Rule
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 4 – Slide 3 of 14
Chapter 5 – Section 4
● For “or” probabilities
 The Addition Rule applies to two disjoint events … the
“easy” case
 The General Addition Rule applies to any two events
● For “and” probabilities
 The Multiplication Rule applies to two independent
events … the “easy” case
 The General Multiplication Rule, this section, applies
to any two events
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 4 – Slide 4 of 14
Chapter 5 – Section 4
● Example




Choosing cards from a deck of cards
E = we chose a diamond as the first card
We did not replace our first card
F = we chose a heart as the second card
● The probability of F happening, given that E has
already happened, is 13/51
 There are 51 cards remaining
 13 of them are hearts
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 4 – Slide 5 of 14
Chapter 5 – Section 4
● 13/51 is called a conditional probability
● The probability of choosing a heart is 13/52
● The probability of choosing a heart, given that
we had already chosen a diamond, is 13/51
● This can be written
P(Heart | Diamond) = 13/51
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 4 – Slide 6 of 14
Chapter 5 – Section 4
● The notation for conditional probability
P(F|E)
is the probability of F given event E
● Only the outcomes contained in the event E are
included in computing conditional probabilities
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 4 – Slide 7 of 14
Chapter 5 – Section 4
● Example
● A group of adults are as per the following table
Right handed
Left handed
Total
Male
38
12
50
Female
42
8
50
Total
80
20
100
● We choose a person at random out of this group
● If E = “male” and F = “left handed”, compute
P(F) and P(F|E)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 4 – Slide 8 of 14
Chapter 5 – Section 4
Right handed
Left handed
Total
Male
38
12
50
Female
42
8
50
Total
80
20
100
● F = “left handed” … P(F) = 20/100 = 0. 20
● E = “male” … P(F|E) = probability of left handed,
given male = 12/50 = 0.24
 There are 50 males and 12 of them are left handed
 The probability of left handed, given male, is 12/50
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 4 – Slide 9 of 14
Chapter 5 – Section 4
● The Conditional Probability Rule is
P
(
E
and
F
)
P
(
F
|
E
)

P
(
E
)
● An interpretation of this is that we only consider
the cases when E occurs (i.e. P(E)), and out of
those, we consider the cases when F occurs (i.e.
P(E and F), since E always has to occur)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 4 – Slide 10 of 14
Chapter 5 – Section 4
● Learning objectives
1 Compute conditional probabilities
2 Compute probabilities using the General Multiplication
Rule
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 4 – Slide 11 of 14
Chapter 5 – Section 4
● We can take the Conditional Probability Rule
P
(
E
and
F
)
P
(
F
|
E
)

P
(
E
)
and rearrange it to be
•
P
(
E
and
F
)

P
(
E
)
P
(
F
|
E
)
● This is the General Multiplication Rule
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 4 – Slide 12 of 14
Chapter 5 – Section 4
● Example
● For a student in a statistics class
 E = “did not do the homework” with P(E) = 0.2
 F = “the professor asks that student a question about
the homework” with P(F|E) = .9
● What is the probability that the student did not
do the homework and the professor asks that
student a question about the homework?
P(E and F) = P(E) • P(F|E) = 0.2 • 0.9 = 0.18
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 4 – Slide 13 of 14
Summary: Chapter 5 – Section 4
● Conditional probabilities P(F|E) represent the
chance that F occurs, given that E occurs also
● The General Multiplication Rule applies to “and”
problems for all events and involves conditional
probabilities
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 4 – Slide 14 of 14
Example
● The following table gives the number (in millions) of men and women
over the age of 24 at each level of educational attainment. (Source:
U.S. Department of Commerce, Census Bureau, Current Population
Survey, March 2004.)
 a. What is the probability that a randomly selected female over the
age of 24 is a college graduate?
 b. Among college graduates over the age of 24, what proportion
are females?
 c. Are the events “college graduate” and “female” independent?
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 4 – Slide 15 of 14
● (0.3535)
● (0.5096)
● (No. The proportion of females is 0.5206, which
is not equal to 0.5096—computed in part b.)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 4 – Slide 16 of 14
Example
● What is the probability that a randomly selected junior is a prospective
donor?
● Among prospective donors, what proportion are juniors?
● Are the events “prospective donor” and “junior” independent?
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 4 – Slide 17 of 14
● (0.5833)
● (0.2500)
● (No, the proportion of juniors is 0.2571, which
does not equal the answer to part b.)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 4 – Slide 18 of 14