creating a 2:1 shot with a fair coin

[Ed Miller]

but again since its between two people it is a finite number of flips, unless you know how to add infinity to your lifespan.

You don't need to live forever if you can learn to flip really freaking fast. [img]/forums/images/icons/smile.gif[/img]

[BruceZ]

I said one thing that isn't right. The average length of this game isn't infinite. The average length of the game is 2 flips. The average length of the game would be infinite if one opponent started with an infinite number of coins (so he could never go broke). That is the classical result I was thinking of, and I didn't bother to do the math for this case. I was under the influence of powerful opiates (legal ones). For this case, the probability is 1/2 that it will end in 1 flip, 1/4 that it will end in 2 flips (guy with 2 coins loses twice), 1/8 that it will end in 3 flips (LWW), etc. So the average length of the game is 1(1/2) + 2(1/2)^2 + 3(1/2)^3 + ... = 2.

It's still true that there is a possibility of the game not ending for any finite number of flips, which was the main point.

If they agree in advance to quit after a certain number of flips, it will still be 2:1 if they quit after an even number of flips, but now there will be the possibility of a tie, so it's really 2:1:t where t is some fractional number of ties. If they quit after an odd number of flips, then the guy that starts with more coins will be better than 2:1 since they are quitting when the other guy is ahead, and he would have won most of these if the game had continued. If they quit after an odd number, then it is back where it started, so it would still be 2:1 if they continued.

[BruceZ]

If they quit after an odd number, then it is back where it started, so it would still be 2:1 if they continued.

Make that an even number. So you want to quit after an even number for 2:1.

[BruceZ]

This will illustrate the difference between a probability of 1 and a dead certainty that someone was asking about earlier. If we can flip as fast as we please, we can flip an infinite number of times in 1 minute. We'll just take 30 seconds for the 1st flip, 15 seconds for the second flip, 7.5 seconds for the 3rd flip, etc. After 1 minute we will have flipped an infinite number of times, or else the game will have ended at some point. It is still possible that the game never ended, and the probability of that it never ended is 0. That is, for each flip we can assign a winner and a loser so that the game doesn't end. There is only 1 way to do this, that is the winners and losers must alternate. Since there are an infinite number of ways the flips can occur, the probability that the game never ends is 0. That is, the probability that the game does end is 1. But it is not a dead certainty that it will end, because there is 1 way that it will not end.