Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 788 — #796
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Chapter 18
Conditional Probability
(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 diagnosed
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 18.20.
Suppose that Let’s Make a Deal is 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 contestant
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 outcomes
and their probabilities you use to model this situation. What is the contestant’s
probability of winning if he uses the best strategy?
(b) If the contestant asks “is the red 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 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 18.21.
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
are two children in every household, that a random child is equally likely to be a
girl or a boy, and that the two children in a household are equally likely to be the
one that opens the door.
A sample space for this experiment has outcomes that are triples whose first
element is either B or G for the sex of the elder child, whose second element is
either B or G for the sex of the younger child, and whose third element is E or Y
indicating whether the elder child or younger child opened the door. For example,
.B; G; Y/ is the outcome that the elder child is a boy, the younger child is a girl, and
the girl opened the door.