How much would you pay for a 1/2 chance to earn $2? (Ignore taxes and transaction costs)
How much would you pay for a 1/4 chance to earn $4?
How much would you pay for a 1/8 chance to earn $8?
How much would you pay for a 1/16 chance to earn $16?
....
How much would you pay for a 1/2^10 chance to earn $2^10? (2^10 is about 1,000)
How much would you pay for a 1/2^20 chance to earn $2^20? (2^20 is about 1 million)
How much would you pay for a 1/2^30 chance to earn $2^30? (2^30 is about 1.1 billion)
How much would you pay for a 1/2^40 chance to earn $2^40? (2^40 is about 1.1 trillion)
Consider each of the 40 similar bets from 2 to 2^40. Suppose you could take them all at the same time in a mutually exclusive game, such that the chance of winning nothing is 1/2^40. For example, the game could be a sequential coin flip until the coin shows tails, and paying 2^n when the nth flip is the first one to show tails, and paying $0 if the all of the first 40 flips are heads.
Is the amount you would pay to play this game equal to the sum of the amounts you would pay for each of the individual games listed above? If not, why not?
Why would you not be willing to pay $40 to play this game?
A variation follows below the fold...
