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Math 160 - Cooley
Intro to Statistics
OCC
Section 4.5 – Conditional Probability
Conditional Probability
The probability that event B occurs given that event A occurs is called a conditional probability. It is
denoted P( B | A) , which is read “the probability of B given A.” We call A the given event.
The Conditional Probability Rule
If A and B are any two events with P(A) > 0, then
P( B | A) 
P( A & B)
P( A)
 Example:
Suppose you roll a single die. What is the probability that you roll a '6', given that the outcome is even?
Solution #1  The sample space for rolling a single die is {1, 2, 3, 4, 5, 6}, a total of 6 outcomes. Since it is
given that the outcome is even, then the revised sample space is now {2, 4, 6}. So, there are
1
now 3 outcomes, one of which is the value ‘6’. Thus, the probability is .
3
Solution #2  Let B = the event that the outcome is a ’6’.
Let A = the event that the outcome is even. Then, P( A) 
Then, P( A & B) 
1
.
2
1
1
, since the probability of a roll of ‘6’ and ‘even’ is .
6
6
Using the formula in the conditional probability rule,
1
P( A & B) 6 1
P( B | A) 
 
1 3
P( A)
2
1
So, the probability using the rule is again .
3
 Example:
Suppose you roll two standard dice. What is the probability that you roll a sum of seven, given that the sum is
not twelve?
Solution 
There are 36 different outcomes for rolling a two dice. Since it is given that the outcome is
not a ‘12’, then, there is only one outcome where a ’12’ is possible. So, the revised sample
space is reduced to 35 different outcomes. There are still 6 different ways to obtain a ‘7’. Thus,
6
the probability is
.
35
Using the formula in the conditional probability rule,
6
P( A & B) 36 6
P( B | A) 


35 35
P( A)
36
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Math 160 - Cooley
Intro to Statistics
OCC
Section 4.5 – Conditional Probability
 Exercises:
1)
Suppose you roll a single die. What is the probability that you roll a '4', given that the outcome is
not a ‘5’?
2)
Suppose you roll two standard dice. What is the probability that you roll a sum of two, given that the
sum is not seven?
3)
An urn contains 5 red marbles and 5 blue marbles. A marble is drawn without replacement. A second
marble is then drawn. What is the probability that you draw a red marble on the second draw, given that
a red marble was obtained on the first draw?
4)
Suppose you flip a fair coin three times. What is the probability that you get exactly three heads
given that at least two heads are showing?
5)
Suppose you draw a card from a standard deck. What is the probability that you draw a heart,
given that a club is not drawn?
6)
Suppose you draw a card from a standard deck. What is the probability that you draw a king, given
that the card is not a 4 or a 6?
7)
Suppose you draw a card from a standard deck. What is the probability that you draw a red card,
given that the card is not a club?
8)
Suppose you draw a card from a standard deck. What is the probability that you draw a diamond,
given that the card is a black card?
9)
Suppose you draw a card from a standard deck. What is the probability that you draw an evennumbered card, given that the card is not an ace or a face card?
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Math 160 - Cooley
Intro to Statistics
OCC
Section 4.5 – Conditional Probability
 Exercises:
The International Shark Attack File, maintained by the American Elasmobranch Society and the Florida
Museum of Natural History, is a compilation of all known shark attacks around the globe from the mid1500s to the present. Following is a contingency table providing a cross-classification of worldwide
reported shark attacks during the 1990s, by country and lethality of attack.
Lethality
Country
10)
Australia
C1
Brazil
C2
South Africa
C3
United States
C4
Other
C5
Total
Fatal
L1
Nonfatal
L2
Total
9
56
65
12
21
33
8
57
65
5
244
249
36
92
128
70
470
540
For a randomly selected attack:
a)
What is the probability that the attack was from Brazil?
b)
What is the probability that the attack was fatal?
c)
What is the probability that the attack was from Brazil and fatal?
d)
What is the probability that the attack was from Brazil, given that the attack was fatal?
e)
What is the probability that the attack was fatal, given that the attack was from Brazil?
f)
What is the probability that the attack was from Australia, given that the attack was nonfatal?
g)
What is the probability that the attack was not from the United States, given that the attack was
fatal?
h)
What is the probability that the attack was fatal, given that the attack was from another country?
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