Exercise 1.56

We are interested in two outcomes of the samme sample space. This is, $S=\{(a_1,a_2,a_3):a_i\in \{1,2,3,...,365\}\}$ The first outcome is $(1,1,1)$, and the second outcome is $(1,2,3)$. The answer follows, since every outcome of the sample space is equally likely.

If the first toss is $T$, Martin can never win, since as soon as $H$ is seen on any subsequent toss, the game stops, and Gale is awarded the win.

If the first toss is $H$, then if the second toss is also $H$, Martin wins. Otherwise, if the second toss is $T$, Gale wins, since as soon as a subsequent toss shows $H$, Gale is awarded a win.

Thus, Martin loses $\frac{3}{4}$ of the time.