One million prisoners

I was hoping it wouldn't have to come to this, but since the correct solution to the 3 prisoners problem is still not being accpeted, let us analyze this extreme case...

There are One Million prisoners in a prison. Let’s call them A_1 through A_(One Million). Tomorrow, 999,999 of them will be executed, but the prisoners don’t know which of them have been chosen. Prisoner A_1 reasons that his chances of survival are 1/(One Million). He then gets bored in his cell and goes to chat with a guard. “Tell me”, he says, “will I be executed tomorrow?” “I’m sorry,” the guard replies, “I am not allowed to tell you that.” “OK”, says Prisoner A_1, “you needn’t tell me anything about myself. But 999,998 of the other 999,999 are sure to be executed. Perhaps all of them but, in any case, at least 999,998 of them.” “That’s right”, confirms the guard. “Very well”, says Prisoner A_1, “tell me the names of 999,998 people who are to be executed.” “All right”, answers the guard, “a_2 through A_999,999 will be executed tomorrow.” On hearing this news Prisoner A_1 feels much more cheerful. He reasons that tomorrow either he or Prisoner A_(One Million) will be executed. And that therefore his chances of survival have increased from 1/(One Million) to 1/2. Is he right?

I 100% agree. It is not 1/2. It is probably still 1 in a million.

Originally posted by Jezz
I 100% agree. It is not 1/2. It is probably still 1 in a million.

No, the survival chance for A_1 is certainly 1 in a million.

Originally posted by DoctorScribbles
I was hoping it wouldn't have to come to this, but since the correct solution to the 3 prisoners problem is still not being accpeted, let us analyze this extreme case...

There are One Million prisoners in a prison. Let’s call th ...[text shortened]... survival have increased from 1/(One Million) to 1/2. Is he right?

Nope - the guard provides no useful information to A_1. A completely different case would be if the guard declared the entire list of those to be executed, except for one omission made uniformly at random (ie he is quite prepared to tell A_1 of his fate, if A_1 were to be executed). In this case A_1 would know, if he wasn't on the list, that either he was the random omission (a priori there's a 1 in a million chance of this happening) or that he is to be spared (ditto). So by Bayes' theorem, A_1's chances of survival would improve to 1/2. Maybe that's what's confusing people.