Section 4.6 Probability
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89. Prove Bayes’ theorem. The proof parallels what was done in Example 73. Let E 1 , …, E n be disjoint events
from a sample space S whose union equals S. If F is another event from S, then Bayes’ theorem says that
the probability of event E i , 1 ≤ i ≤ n, given event F, is
P(E i 0 F ) =
P(F 0 E i )P(E i )
∙
n
k=1
P(F 0 E k )P(E k )
a. Use the definition of P(E i 0 F ) and P(F 0 E i ) to prove that
P(E i 0 F ) =
P(F 0 E i )P(E i )
P(F )
b. Prove that
P(F ) = ∙
n
k=1
P(F d E k )
c. Use the definition of P(F 0 E i ) and the result from part (b) to prove that
P(F ) = ∙
n
k=1
P(F 0 E i )P(E i )
d. Using parts (a) and (c), prove Bayes’ theorem.
90. Toys for boys and for girls are donated to a benefit event by two groups. The Lakeville Do-Gooders
donated 5 toys for boys and 7 for girls. The Southside Champions Club donated 6 toys for boys and 5 for
girls. The master of ceremonies pulls the first toy out of a bin and it’s a toy for a boy. Find the probability
that it was donated by the Do-Gooders.
91. An online pharmacy sells an over-the-counter drug, medication X, that is used for a variety of purposes.
The pharmacy has data that say that 18% of its customers are HIV positive, 9% of its HIV-positive customers buy medication X, and 3% of the customers who are not HIV positive buy medication X. Find the
probability that a customer who buys medication X is HIV positive. The pharmacy can use this data to
shape its marketing/advertising plan.
92. Of the high blood pressure patients in a particular clinic, 62% are treated with medication X, the remainder
with medication Y. It is known that 1.4% of patients using medication X suffer from fainting spells, as do
2.9% of the patients using medication Y. A patient known by the clinic to have high blood pressure suffers
a fainting spell, but she does not remember which medication she is on. Which medication is she most
likely to be taking? (Hint: Let E 1 and E 2 —treated with X and treated with Y, respectively—be events in
the sample space of all patients with high blood pressure in the clinic, and let F be the event of fainting;
find P(X 0 F ) and P(Y 0 F ).)
93. a. A fair die is rolled once. Let the random variable X equal the value that comes up. Find the expected
value of X, E(X ).
b. The die is now “loaded” so that a 2 comes up twice as often as any other number. Find the new expected
value of X.
c. Your answer to part (b) should be (greater than, less than) your answer to part (a). Explain why.
321
89. Prove Bayes’ theorem. The proof parallels what was done in Example 73. Let E 1 , …, E n be disjoint events
from a sample space S whose union equals S. If F is another event from S, then Bayes’ theorem says that
the probability of event E i , 1 ≤ i ≤ n, given event F, is
P(E i 0 F ) =
P(F 0 E i )P(E i )
∙
n
k=1
P(F 0 E k )P(E k )
a. Use the definition of P(E i 0 F ) and P(F 0 E i ) to prove that
P(E i 0 F ) =
P(F 0 E i )P(E i )
P(F )
b. Prove that
P(F ) = ∙
n
k=1
P(F d E k )
c. Use the definition of P(F 0 E i ) and the result from part (b) to prove that
P(F ) = ∙
n
k=1
P(F 0 E i )P(E i )
d. Using parts (a) and (c), prove Bayes’ theorem.
90. Toys for boys and for girls are donated to a benefit event by two groups. The Lakeville Do-Gooders
donated 5 toys for boys and 7 for girls. The Southside Champions Club donated 6 toys for boys and 5 for
girls. The master of ceremonies pulls the first toy out of a bin and it’s a toy for a boy. Find the probability
that it was donated by the Do-Gooders.
91. An online pharmacy sells an over-the-counter drug, medication X, that is used for a variety of purposes.
The pharmacy has data that say that 18% of its customers are HIV positive, 9% of its HIV-positive customers buy medication X, and 3% of the customers who are not HIV positive buy medication X. Find the
probability that a customer who buys medication X is HIV positive. The pharmacy can use this data to
shape its marketing/advertising plan.
92. Of the high blood pressure patients in a particular clinic, 62% are treated with medication X, the remainder
with medication Y. It is known that 1.4% of patients using medication X suffer from fainting spells, as do
2.9% of the patients using medication Y. A patient known by the clinic to have high blood pressure suffers
a fainting spell, but she does not remember which medication she is on. Which medication is she most
likely to be taking? (Hint: Let E 1 and E 2 —treated with X and treated with Y, respectively—be events in
the sample space of all patients with high blood pressure in the clinic, and let F be the event of fainting;
find P(X 0 F ) and P(Y 0 F ).)
93. a. A fair die is rolled once. Let the random variable X equal the value that comes up. Find the expected
value of X, E(X ).
b. The die is now “loaded” so that a 2 comes up twice as often as any other number. Find the new expected
value of X.
c. Your answer to part (b) should be (greater than, less than) your answer to part (a). Explain why.