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Exercise 2.45

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(a)

Since the prizes are independent for each door, and since the strategy is switch doors every time, what is behind Door

1

is irrelevant.

Possible outcomes for doors $2$ and $3$ are Goat and Car with probability $2 pq$ , in which case the contestant wins, Car and Car with probability $p^{2}$ , in which case the contestant wins again, and Goat and Goat with probability $q^{2}$ , in which case the contestant loses.

Thus,

P (S) = \frac{p^{2} + 2 pq}{p^{2} + 2 pq + q^{2}} = \frac{p^{2} + 2 pq}{{(p + q)}^{2}} = p^{2} + 2 pq .

(b)

There are two scenarios in which Monty opens Door

2

. Either Door

3

contains a Car and Door

2

contains a Goat, which happens with probability

pq

, or both doors contain Goats and Monty randomly chooses to open Door

2

, which happens with probability

\frac{1}{2} q^{2}

. Contestant wins in the first case and loses in the second case.

Thus,

P (S | D_{2}, G_{2}) = \frac{pq}{pq + \frac{1}{2} q^{2}} .

fifthist

2021-12-05 00:00

Exercise 2.45

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