2.46 Suppose an individual is randomly selected from the population of all adult males living in the United States. Let A be the event that the selected individual is over 6 ft in height, and let B be the event that the selected individual is a professional basketball player. Which do you think is larger, P(A|B) or P(B|A)? Why?
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2.50 A department store sells sport shirts in three sizes (small, medium, and large), three patterns (plaid, print, and stripe), and two sleeve lengths (long and short). The accompanying tables give the proportions of shirts sold in the various category combinations. a. What is the probability that the next shirt sold is a medium, long-sleeved, print shirt? b. What is the probability that the next shirt sold is a medium print shirt? c. What is the probability that the next shirt sold is a short- sleeved shirt? A long-sleeved shirt? d. What is the probability that the size of the next shirt sold is medium? That the pattern of the next shirt sold is a print? e. Given that the shirt just sold was a short-sleeved plaid, what is the probability that its size was medium? Short-sleeved Pattern Size Pl Pr S .04 .02 M .08 .07 L .03 .07 Long-sleeved Pattern Size Pl Pr S .03 .02 M .10 .05 L .04 .02
St
.05
.12
.08
St
.03
.07
.08
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A set of events A
1
, A
2
, …, A
k
is
exhaustive
if at least one
of the events must occur.
Total-Probability Rule
Let A
1
, A
2
, …, A
k
be mutually exclusive and exhaustive
events.
The unconditional probability B, P(B), can then
be written as a weighted average of the conditional
probabilities of B given A
i
, P(B | A
i
), as follows:
P(B) =
P(B | A
i
)
x
P(A
i
)
Bayes’ Rule
If A and B are any events whose probabilities are not 0 or
1,
P(A|B) =
P(B|A)P(A)
P(B|A)P(A) + P(B|A')P(A')
