- #1
pig
- 94
- 0
The thread about Zeno reminded me of a paradox that I was never able to understand.
Let's say you play a game, where you repeatedly flip a coin, and the first time the coin lands tails, you stop flipping. Then you get 2^n coins, where n is the number of heads flips. How much money is the game worth?
The probability of n being 0 is 1/2 (first flip tails).
The probability of n being 1 is 1/2*1/2 (first flip heads, second flip tails).
The probability of n being 2 is 1/2*1/2*1/2 (first and second flip heads, third tails).
And so on.
So, the expected payoff is:
(1/2)^1 * 2^0 + (1/2)^2 * 2^1 + (1/2)^3 * 2^2 + ...
= 1/2 + 1/2 + 1/2 + ...
The expected payoff is infinite, and therefore I should offer any finite amount of money to play this game. Can anyone explain?
Either intuition and a finite number of experiments don't work well with an infinite number of possibilities, or math doesn't.
Let's say you play a game, where you repeatedly flip a coin, and the first time the coin lands tails, you stop flipping. Then you get 2^n coins, where n is the number of heads flips. How much money is the game worth?
The probability of n being 0 is 1/2 (first flip tails).
The probability of n being 1 is 1/2*1/2 (first flip heads, second flip tails).
The probability of n being 2 is 1/2*1/2*1/2 (first and second flip heads, third tails).
And so on.
So, the expected payoff is:
(1/2)^1 * 2^0 + (1/2)^2 * 2^1 + (1/2)^3 * 2^2 + ...
= 1/2 + 1/2 + 1/2 + ...
The expected payoff is infinite, and therefore I should offer any finite amount of money to play this game. Can anyone explain?
Either intuition and a finite number of experiments don't work well with an infinite number of possibilities, or math doesn't.