z
To determine the probability of 2 white shirts being selected we use formula:
P(AB) = P(A) P(BA)
P
(
W
1
and
W
2
) =
P
(
W
1
)
P
(
W
2

W
1
)
= (9/12)(8/11) = 0.55
General Multiplication Rule  Example
A golfer has 12 golf shirts in his closet.
Suppose 9 of these shirts are white and
the others blue. He gets dressed in the
dark, so he just grabs a shirt and puts it
on. He plays golf two days in a row and
does not do laundry.
What is the likelihood both shirts selected
are white?
157
Exercises 162:28
z
3 Defective and 17 Good Toothbrushes shipped
z
(a) Probability 1st 2 are Defective?
z
P(Defective #1) * P(Defective #2/Defective #1)
z
3/20 * 2/19 = 6/380 = 0.01579
z
(b) Probability NONE of 1st 2 are Defective?
z
P(Good #1) * P(Good #2/Good #1)
z
17/20 * 16/19 = 272/380 = 0.7158
Exercises 163:32
z
3 Strangers, questions about their birthdays
z
(a) Probability ALL born Wednesday?
z
1/7 * 1/7 * 1/7 = 0.002915
z
(a) Probability NONE born Saturday?
z
6/7 * 6/7 * 6/7 = 0.6297
z
(c) Probability born on DIFFERENT days?
z
7/7 * 6/7 * 5/7 = 0.6122
Contingency Tables
A CONTINGENCY TABLE
is a table used to classify sample
observations according to two or more identifiable characteristics
E.g. A survey of 150 adults classified each as to gender and the number of
movies attended last month. Each respondent is classified according to
two criteria—the number of movies attended and gender.
158
Contingency Tables  Example
A sample of executives were surveyed about their loyalty to their company.
One of the questions was, “If you were given an offer by another
company equal to or slightly better than your present position, would
you remain with the company or take the other position?”
The
responses of the 200 executives in the survey were crossclassified
with their length of service with the company.
What is the probability of randomly selecting an executive who is loyal to
the company (would remain) and who has more than 10 years of
service?
159
Event
A
1
happens if a randomly selected executive will remain with the
company despite an equal or slightly better offer from another company.
Since there are 120 executives out of the 200 in the survey who would
remain with the company
P
(
A
1
) = 120/200, or .60.
Event
B
4
happens if a randomly selected executive has more than 10 years
of service with the company. Thus, P(B
4
 A
1
) is the conditional
probability that an executive with more than 10 years of service would
remain with the company. Of the 120 executives who would remain 75
have more than 10 years of service, so
P(B
4
 A
1
)
= 75/120
.
Contingency Tables  Example
159
Tree Diagrams
A
tree diagram
is useful for portraying
conditional and joint probabilities.
It is
particularly useful for analyzing business
decisions involving several stages.
A
tree diagram
is a graph that is helpful in
organizing calculations that involve several
stages. Each segment in the tree is one stage of
the problem. The branches of a tree diagram are
weighted by probabilities.
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 Spring '11
 Leany
 Conditional Probability, Probability, Survey of Probability Concepts