Imagine a game that is played by flipping a fair coin until it comes up tails, and the total number of flips, n, determines the prize, which equals $2n. Thus if the coin comes up tails the first time, the prize is $2^1 = $2, and the game ends. If the coin comes up heads the first time, it is flipped again. If it comes up tails the second time, the prize is $2^2, = $4, and the game ends. If it comes up heads the second time, it is flipped again. And so on. There are an infinite number of possible ‘consequences’ (runs of heads followed by one tail) possible.
In other words you have a 1 in $2 chance of getting $2 plus a one in 4 chance of getting $4 on your last flip and a one in eight chance of getting $8 a one in 16 chance of getting $16 and so on. If you won 3 times you would get $2 for the first win, $4 for the 2nd and $8 for the third, or a total of $14. The amount you can win accelerates exponetially.
Theoretically you could win any amount of money including more dollars then there are subatomic particles in the universe. (If the coin came up tails often enough which is very unlikely) Of course in the real world the amount of money is not infinite and also if you dumped a lot of new money in to the economy you would get inflation and the value of the money would be lower. For the purpose of this question you can assume that the money reflects actual new resources and there is no inflation.
How much is the maximum amount you would pay to be able to participate in this game if there was a limit of 25 coin flips (a total of almost $70mil in possible winnings), if there was a limit of 40 coin flips (about 1.1 trillion dollars on the last flip or $2.2tril total) and if there was no limit (and thus no limit on your potential winnings)?
Tim |
