so this is what you're trying to tell me...

Tom's probability were really between himself and a collective known as 'other'

other contained 2 people so the probability it was between any peson would be 1/3

so the probability it was tom is 1/3 and the probability it was other is 2/3.

Also the number of people reducing other doesn't change 'other's' its overall probability.

Exactly.

the problem with this is looking at it from an inside point of view and an outside point of view...

because quite clearly from the other man's point of view, he would also stand a 1/3 of chance of survival from his pov while tom would now have a 2/3 probability in being killed.

Again, that's right. (That's why this is the problem my old professor used to show us how bad our math intuition is.) From Harry's point of view (assuming he was also told), Dick is going to live, so there is a 1/3 chance he'll die, and a 2/3 chance Tom will die. All at the same time that Tom sees he has a 1/3 chance and Harry has a 2/3. And that's where human intuition has the problem. But probability is all about your point of view and when things happen in relation to other events.

it also makes a difference whether the random selection was done before or after the knowledge that one of them was or was not to be killed because tom's perception of having a 1/3 chance may have not been accurate from the get go.This quite clearly changes things.

True. If the guards had decided Dick would live before making their choice, Tom's survival is 50-50 because there were only two options to choose from. But because the guards didn't know Dick would live until

*after *they made their choice, they each have a 1/3 chance because there were three options to choose from. And eliminating an option after the fact doesn't change that.

I'll accept their answer and the logic, but personally i don't think its a practical approach.

it's like the two finalists in pop idol instead of having a 1/2 chance of winning actually have a 1/5000000000 from the entire population/people that applied for the contest. and they both think the other has a 4999999999999/500000000000 chance of becoming the winner

That's a different situation. If you tried to pick the idol winner before any eliminations (assuming equal quality of performance), then it would be 1/5000000000 or whatever.

But after each round of eliminations, the audience gets to repick from the leftovers. So the next time it becomes 1/50000. And then 1/12. And eventually 1/2 (or whatever the numbers are). If the audience never got to change their vote before the first round of eliminations, then the chance any particular vote was right would remain 1/5000000000 throughout the competition.

If you want physical proof, you can try this experiment yourself. Get three playing cards; Ace, King, Queen. Put them upside down, mix them up, and always try to pick the Ace. Now, since any card is just as good as another when they're upside down, just always pick the one on the left as yours. Go ahead and flip up the one on the right side. It's either an Ace, or not (2/3 chance it won't be). No matter what comes up, though, it doesn't change what the left card (your card) will be. So go ahead and flip your card up. If you keep track, your card (the left card) will be an Ace 1/3 of the time, even though you eliminate one of the other options before turning up your card.

Now, change it around a bit. Flip up the right-side card. If it's the Ace, redo it. If it's not the Ace, throw it out, mix up the two remaining cards and repick one as yours. Now there will be a 50-50 chance of getting it right because you're basically rerunning the experiment with only two options. If you keep track this time (mix, flip right-side, mix again, then choose), you'll hit the Ace half the time.