Mathematics for Computer Science

Chapter 17 Conditional Probability728

GP:The event that the contestantguesses the door with theprize behind it on his
first guess.

OP:The event that the game is restarted at least once. Another way to describe
this is as the event that the door Carol firstopens has aprize behind it.

Give the values of the following probabilities:

(a)PrŒGPç

(b)Pr



OP

ˇˇ

GP



(c)PrŒOPç

(d)the probability that the game will continue forever

(e)When Carol finally picks the goat, the contestant has the choice of sticking or
switching. Let’s say that the contestant adopts the strategy of sticking. LetW be
the event that the contestant wins with this strategy, and letwWWDPrŒW ç. Express
the following conditional probabilities as simple closed forms in terms ofw.

i) Pr



W jGP



ii) Pr



W

ˇˇ

GP\OP



iii) Pr



W

ˇˇ

GP\OP



D

(f)What is the value of PrŒW ç?

(g)For any final outcome where the contestant wins with a “stick” strategy, he
would lose if he had used a “switch” strategy, and vice versa. In the original Monty
Hall game, we concluded immediately that the probability that he would win with
a “switch” strategy was 1 PrŒW ç. Why isn’t this conclusion quite as obvious for
this new, restartable game? Is this conclusion still sound? Briefly explain.

Problem 17.16.
There are two decks of cards, the red deck and the blue deck. They differ slightly
in a way that makes drawing the eight of hearts slightly more likely from the red
deck than from the blue deck.
One of the decks is randomly chosen and hidden in a box. You reach in the
box and randomly pick a card that turns out to be the eight of hearts. You believe
intuitively that this makes the red deck more likely to be in the box than the blue
deck.