>>>You pay me a certain amount of money at the beginning. I flip a coin. If "tail" appears, the game ends - you get nothing in exchange for you initial bet. If "head" appears, I pay you $1, and the game continues. If "head" appears for the second time,I pay you $2. The third time, $4, the fourth time $8, etc. - but as soon as "tail" appears, the game is over.
>>>
>>>How much would you be willing to pay to play this game?
>>>
>>>Hilmar.
>>
>>I think I hadn't made it clear that the payments are additive. For instance, if "head" appears thrice, you get $1 + $2 + $4 = $7.
>>
>>Well, since nobody responded in many days anyway, I will give an analysis. The probability of getting "head" the first time is 1/2; making a value of 1/2 * $0.50. For the second "head", the probability is 1/4, the total value is 1/4 * $2 = $0.50. For the third head, the total value is 1/8 * $4 = $0.50.
>>
>>The total value of the game, in dollars, is 1/2 + 1/2 + 1/2 ... that is, infinite.
>>
>>Of course, playing the game a single time, you will get a finite amount of money. It is only the average that is infinite. Strange, huh?
>>
>>Hilmar.
>I'm not sure the analysis is totally correct, but agree that the possible winnings is theorically infinite, but gets there less rapidly - my analysis is
>
>50% chance losing everything
>25% chance winning $1
>12.5% chance winning $3
>6.25% chance winning $7

Well, and how much does this add up to? (25% * $1 + 12.5% * $3, etc.).

These terms should be added to get the average value of the game: 0.25 + 0.375 + 0.4375 + 0.4688 + 0.4844 + 0.4922 + 0.4961 ... (the terms rapidly approach 0.5).

>Betting $1 would give me a 50% chance of winning at least $1, so in the long run I'm likely to profit. 50% of the time I lose, 25% of the time break even, 25% of the time profit by at least $2, so the odds are in my favour.
>
>If I bet $3, I would only have a 25% chance of winning at least $3, 50% of the time I would lose $3, 25% of the time I would lose $2, 12.5% of the time I would break even & only 12.5% of the time would I be in profit. At this level the odds are very much in your favour.

It seems to me that your are confusing "average" with "median" here. If you get less than $3 in half the games, but much more than $6 on the other half, you are still in a favourable position. The game is worth more than $3, on the average.

>Looking at it from that point of view, I would ask you - what is the minimum bet you would accept from a person before you agreed to play your game.

I wouldn't accept this bet (on the "house" side) under any circumstances.

Hilmar.