Mathematics for Computer Science

17.8. Mutual Independence 727

(a)If you have one ace, what is the probability that you have a second ace?

(b)If you have the ace of spades, what is the probability that you have a second
ace? Remarkably, the answer is different from part (a).

Problem 17.14.
Suppose PrŒ
çWS!Œ0;1çis a probability function on a sample space,S, and letB
be an event such that PrŒBç > 0. Define a function PrBŒ
çon outcomes! 2 Sby
the rule:

PrBŒ!çWWD

(

PrŒ!ç=PrŒBç if! 2 B;

0 if!...B:

(17.7)

(a)Prove that PrBŒ
çis also a probability function onSaccording to Defini-
tion 16.5.2.

(b)Prove that

PrBŒAçD

PrŒA\Bç

PrŒBç

for allAS.

(c)Explain why the Disjoint Sum Rule carries over for conditional probabilities,
namely,

Pr



C[DjB



DPr



C jB



CPr



DjB



.C;Ddisjoint/:

Give examples of several further such rules.

Exam Problems

Problem 17.15.
Here’s a variation of Monty Hall’s game: the contestant still picks one of three
doors, with a prize randomly placed behind one door and goats behind the other
two. But now, instead of always opening a door to reveal a goat, Monty instructs
Carol torandomlyopen one of the two doors that the contestant hasn’t picked. This
means she may reveal a goat, or she may reveal the prize. If she reveals the prize,
then the entire game isrestarted, that is, the prize is again randomly placed behind
some door, the contestant again picks a door, and so on until Carol finally picks a
door with a goat behind it. Then the contestant can choose tostickwith his original
choice of door orswitchto the other unopened door. He wins if the prize is behind
the door he finally chooses.
To analyze this setup, we define two events: