AP Statistics Section 6.3 B
Conditional probability
Slim considers himself a pretty good poker
player, at least when he is the only one playing.
He has been dealt 4 cards and wishes to know
the probability that his 5th card will be an ace.
Can we figure this probability?
Not without knowing what the
first four cards were
Find P(5th card is an ace) if his first 4 cards are
two 3s, a 7 and a jack
4
1

48 12
Find P(5th card is an ace) if his first
4 cards are two 3s, a 7 and an ace.
3
1

48 16
The probability we assign to an
event can change if we know that
some other event has occurred.
When a probability is based on the
knowledge of a previous event it is called
conditional probability
The notation for conditional
P( A / B) This
probability is _______.
notation is read:
probability of A given that B has already occurred.
Example: Here is a table of grades awarded at a university by school.
Grade Level
School
A
B
Below B
Total
Liberal Arts
2,142
1,890
2,268
6,300
Engineering
368
432
800
1,600
Health Services
882
630
388
2,100
3,392
2,952
3,656
10,000
Total
Consider the events:
E = grade comes from an Engineering course
B = the grade is a B.
P(B) 
2952
369

10000 1250
P(B/E) 
432
27

1600 100
Example: Here is a table of grades awarded at a university by school.
Grade Level
School
A
B
Below B
Total
Liberal Arts
2,142
1,890
2,268
6,300
Engineering
368
432
800
1,600
Health Services
882
630
388
2,100
3,392
2,952
3,656
10,000
Total
Consider the events:
E = grade comes from an Engineering course
B = the grade is a B.
P(E ) 
1600
4

10000 25
P(E/B) 
432
6

2952 41
Example: Here is a table of grades awarded at a university by school.
Grade Level
School
A
B
Below B
Total
Liberal Arts
2,142
1,890
2,268
6,300
Engineering
368
432
800
1,600
Health Services
882
630
388
2,100
3,392
2,952
3,656
10,000
Total
Consider the events:
E = grade comes from an Engineering course
B = the grade is a B.
1600  2952  432 4120 103


10000
10000 250
432
2

10000 25
P( E  B) 
P(E  B) 
Note: In conditional probability the
condition has the effect of
reducing the size of the sample
space (i.e. the denominator in the
probability fraction)
General Multiplication Rule for
Any Two Events:
P( A  B)  P( A)  P( B / A)
Example: Slim is still at the poker table. Slim sees 11
cards on the table. Of these, 4 are diamonds. What is
the probability of Slim being dealt 2 diamonds from the
deck?
9 8
72
9



41 40 1640 205
If we take the General Multiplication
Rule above and divide both sides by
P(A) we obtain
P( B  A)
P( B / A) 
P( A)
Example: Motor vehicles are classified as either light trucks or
cars and as either domestic or imported. In early 2004, 69% of
vehicles sold were light trucks, 78% were domestic and 55%
were domestic light trucks. Let T be the event a vehicle is a light
truck and D be the event it is domestic. Write each of the
following in terms of events T and D and give the probability.
a. The vehicle is a car
P(T )  1  P(T )  1  .69  .31
c
Example: Motor vehicles are classified as either light trucks or
cars and as either domestic or imported. In early 2004, 69% of
vehicles sold were light trucks, 78% were domestic and 55%
were domestic light trucks. Let T be the event a vehicle is a light
truck and D be the event it is domestic. Write each of the
following in terms of events T and D and give the probability.
b. The vehicle is an imported car
P( D  T )  1  (.14  .55  .23)  .08
c
c
Example: Motor vehicles are classified as either light trucks or
cars and as either domestic or imported. In early 2004, 69% of
vehicles sold were light trucks, 78% were domestic and 55%
were domestic light trucks. Let T be the event a vehicle is a light
truck and D be the event it is domestic. Write each of the
following in terms of events T and D and give the probability.
c. If a vehicle is a car, what is the probability that it is imported?
c
c
P
(
D

T
) .08 8
c
c
P( D / T ) 


c
.31 31
P(T )
Example: Motor vehicles are classified as either light trucks or cars and
as either domestic or imported. In early 2004, 69% of vehicles sold
were light trucks, 78% were domestic and 55% were domestic light
trucks. Let T be the event a vehicle is a light truck and D be the event
it is domestic. Write each of the following in terms of events T and D
and give the probability.
d. Are the events “vehicle is a car” and “vehicle is imported”
independent?
Does P(T c  D c )  P(T c )  P( D c )
.08  .31  .22
.08  .0682
Not Independent
Does P( D c / T c )  P( D c )
8
 .22
31
Not Independent