Here we look at some variations of Example 1.3.4.
(a) In the warden's calculation of Example 1.3.4 it was assumed that if A were to be pardoned, then with equal probability the warden would tell A that either B or C would die. However, this need not be the case. The warden can assign probabilities γ and 1 - γ to these events, as shown here:

Calculate P(A|W) as a function of 7. For what values of 7 is P(A|W) less than, equal to, or greater than 1/3?
(b) Suppose again that γ = 1/2 as in the example. After the warden tells A that B will die, A thinks for a while and realizes that his original calculation was false. However, A then gets a bright idea. A asks the warden if he can swap fates with C. The warden, thinking that no information has been passed, agrees to this. Prove that A's reasoning is now correct and that his probability of survival has jumped to 2/3!
A similar, but somewhat more complicated, problem, the "Monte Hall problem" is discussed by Selvin (1975). The problem in this guise gained a fair amount of notoriety when it appeared in a Sunday magazine (vos Savant 1990) along with a correct answer but with questionable explanation. The ensuing debate was even reported on the front page of the Sunday New York Times (Tierney 1991). A complete and somewhat amusing treatment is given by Morgan et al. (1991) [see also the response by vos Savant 1991]. Chun (1999) pretty much exhausts the problem with a very thorough analysis.

Warden tells A B dies C dies C dies B dies Prisoner pardoned with probability y with probability 1 -y