2.3.5 problem 45

(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 $2\mathrm{pq}$, 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,

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

(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 $\mathrm{pq}$, 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,

$$P\left(S\right|D_2,G_2)=\frac{\mathrm{pq}}{\mathrm{pq}+\frac{1}{2}{q}^2}.$$