Each prisoner had an equal chance of being the one chosen to be executed. So we have three cases:

Prisoner executed: A B C Probability of this case: 1/3 1/3 1/3

Now, if A is to be executed, the warden will randomly choose either B or C, and tell A that name. When B or C is the one to be executed, there is only one prisoner other than A who will not be executed, and the warden will always give that name. So now we have:

Prisoner executed: A A B C Name given to A: B C C B Probability: 1/6 1/6 1/3 1/3

We can calculate all this without knowing the warden's answer. When he tells us B will not be executed, we eliminate the middle two choices above. Now, among the two remaining cases, C is twice as likely as A to be the one executed. Thus, the probability that A will be executed is still 1/3, and C's chances are 2/3.