Death Probability! Really Hard One!

[roolstar]

Alfred, Bob and Charlie (A, B, C) are three prisoners.

One day, one of the guards came and told them: "Tomorrow, 2 among you will be executed, and 1 will be set free. I know who will die myself but I don't want to tell you now, I prefer to tease you during your last night here."

That night while the other prisoners were sleeping, Alfred was unable to: he was calculating the probaility of being executed the next day. According to his calculations, he had a 2/3 (66.67%) probability of being executed tomorrow.

But then "EUREKA"!

He called the guard and said: "I don't want you to tell me if I'm gonna die tomorrow, but can you tell me which among the other two will die?"

The guard answered him: "Mmmm, ok. Charlie will be one of the two prisoners who will get executed tomorrow. But I will not tell you if the second one will be you or Bob!"

Alfred with a smile: "Thank you so much man!'

Guard: "But I don't understand how this will make you feel any better."

Alfred replied with a huge smile: "Well, now I have a 1/2 (or 50%) chance of being executed instead of the 2/3 (66.67%) I had before your answer: either me or Bob will die tomorrow so 1 out of 2!"

Is Alfred right in his reasoning?

[Guest]

If instead we are asking about the probability that Alfred will survive based on the fact that Charlie will die for sure and that one of either Bob or Alfred will die, then he has a 1/2 chance of survival. The question and the setup have changed. Probabilities change when new parameters are given. To say that Bob now has a 2/3 chance and Alfred has a 1/3 chance of survival is mathematical absurdity and it not grounded in reality.

That is what's being asked and the chance of survival for Alfred stays at 2/3 and the reasons for it have been given many times in this thread. Just claiming it's a "mathematical absurdity and it not grounded in reality" and not giving a mathematical reason why is not how things work around here.

Look at Martini's example using the lottery:

"One million people, including yourself, purchase lottery tickets of which there will be three winners. If you could get someone with inside info to rule out 999,994 losers, other than yourself if you should be one, is your probability of being a winner 1/2?"

According to your rationale you would now have a 1/2 probability of winning the lottery. But why should you? All the lottery official did in his example was rule out losers not including yourself. Because you were excluded from the list of possible losers, your chances remain the same- three in a million. You already knew he could rule out 999,994 losers. However, five others are left from all of the ones that were excluded randomly- they have an awesome chance of being winners.

It's the same in this riddle. Alfred already knew that one of the others was going to be executed, so knowing which one it was doesn't help his chances. But Bob wasn't excluded from the selection process. He's in the same boat as the five others in the lottery example. His odds of living just get better (see post #13).

Someone in all this mess got this right, but was shoved aside by the statisticians among you.

Here's the rub: The guard says "I know who will die myself but I don't want to tell you now"

Thus, statistics do not apply to who is going to die, but merely to Al's ability to guess. His knowledge of one of the others being a "chosen" one has absolutely NO bearing on who the two to die are, because this is a predetermined condition that DOES NOT CHANGE.

I'm not sure what you're saying. Are you agreeing that the probability is the same for Alfred but has increased for Bob, or are you saying something else?

[Guest]

I'm saying the probability for all three have never changed, as they are fixed. The guard KNOWS who is going to die. Therefore Al's chances of living CANNOT CHANGE. Only the probability of him guessing who will live changes.

If I know that I have two red marbles in my pocket, but I tell you to guess what color they both are and tell you that they are primary colors, does it change the fact that the two marbles are red? No. Probability only matters when it comes to your guess. You will guess r/r r/b r/y b/b y/y b/y. IF I then tell you one is red, does it change the colors? NO! The probability of the ACTUAL color changing is zero. They have been and always will be red. Only the probability of your guessing correctly changes.

Probability only matters when you have an unknown state and more than one solution. Here, there is ONLY ONE SOLUTION. The guard KNOWS who is living and dying. It's a fixed state with no chance of changing. The only thing that changes is Al's probability of guessing correctly, not the probability he will live or die.

[Guest]

"One million people, including yourself, purchase lottery tickets of which there will be three winners. If you could get someone with inside info to rule out 999,994 losers, other than yourself if you should be one, is your probability of being a winner 1/2?"

According to your rationale you would now have a 1/2 probability of winning the lottery. But why should you? All the lottery official did in his example was rule out losers not including yourself. Because you were excluded from the list of possible losers, your chances remain the same- three in a million. You already knew he could rule out 999,994 losers. However, five others are left from all of the ones that were excluded randomly- they have an awesome chance of being winners.

I think you are missing my point. I believe it still depends upon the question. If the question is, what are your chances of winning the lottery, the answer would be three in a million. Even after the official ruled out the 999,994 losers the answer would still be three in a million. But if the question is, based on the information you now know what are your chances of winning, the answer WOULD be 1/2. The fact that the lottery offcial merely ruled out 999,994 losers did not change anything in reality, but, probability depends on the question you are asking and the parameters that are given. Thus if the question is, what were Alfred's chances of survival based on the initial conditions, the answer is 1/3. If the question is, between him and Bob (now that we know that Charlie will die and that either Bob or Alfred will die) what are his chances of survival, the answer is 1/2. To say that between the two of them one has a higher probabiility than the other, to me, does not make sense. I could be wrong, but to me that just sounds like it's not grounded in reality and that we are just playing with the numbers with no regard to common sense.

Edited April 21, 2008 by baker.uva

[Guest]

I think you are missing my point. I believe it still depends upon the question. If the question is, what are your chances of winning the lottery, the answer would be three in a million. Even after the official ruled out the 999,994 losers the answer would still be three in a million. But if the question is, based on the information you now know what are your chances of winning, the answer WOULD be 1/2. The fact that the lottery offcial merely ruled out 999,994 losers did not change anything in reality, but, probability depends on the question you are asking and the parameters that are given. Thus if the question is, what were Alfred's chances of survival based on the initial conditions, the answer is 1/3. If the question is, between him and Bob (now that we know that Charlie will die and that either Bob or Alfred will die) what are his chances of survival, the answer is 1/2. To say that between the two of them one has a higher probabiility than the other, to me, does not make sense. I could be wrong, but to me that just sounds like it's not grounded in reality and that we are just playing with the numbers with no regard to common sense.

I read up on the Monty Hall problem on answers.com and I am wrong in my reasoning. Alfred has a 1/3 chance of survival and Bob has a 2/3 chance.