17.8. Mutual Independence 731
Suppose there was a vaccine to prevent Beaver Fever, but the vaccine was expen-
sive or slightly risky itself. If you were sure you were going to suffer from Beaver
Fever, getting vaccinated would be worthwhile, but even if Dr. Meyer diagnosed
you as a future sufferer of Beaver Fever, the probability you actually will suffer
Beaver Fever remains low (about 1/32 by part (d)).
In this case, you might sensibly decide not to be vaccinated—after all, Beaver
Fever is notthatbad an affliction. So the diagnostic test serves no purpose in your
case. You may as well not have bothered to get diagnosed. Even so, the test may
be useful:
(e)Suppose Dr. Meyer had enough vaccine to treat 2% of the population. If he
randomly chose people to vaccinate, he could expect to vaccinate only 2% of the
people who needed it. But by testing everyone and only vaccinating those diag-
nosed as future sufferers, he can expect to vaccinate a much larger fraction people
who were going to suffer from Beaver Fever. Estimate this fraction.
Problem 17.19.
Suppose thatLet’s Make a Dealis played according to slightly different rules and
with a red goat and a blue goat. There are three doors, with a prize hidden behind
one of them and the goats behind the others. No doors are opened until the con-
testant makes a final choice to stick or switch. The contestant is allowed to pick a
door and ask a certain question that the host then answers honestly. The contestant
may then stick with their chosen door, or switch to either of the other doors.
(a)If the contestant asks “is there is a goat behind one of the unchosen doors?”
and the host answers “yes,” is the contestant more likely to win the prize if they
stick, switch, or does it not matter? Clearly identify the probability space of out-
comes and their probabilities you use to model this situation. What is the contes-
tant’s probability of winning if he uses the best strategy?
(b)If the contestant asks “is theredgoat behind one of the unchosen doors?” and
the host answers “yes,” is the contestant more likely to win the prize if they stick,
switch, or does it not matter? Clearly identify the probability space of outcomes
and their probabilities you use to model this situation. What is the contestant’s
probability of winning if he uses the best strategy?
Problem 17.20.
You are organizing a neighborhood census and instruct your census takers to knock
on doors and note the sex of any child that answers the knock. Assume that there