Bet You a Million

Good straight-forward logic from Thomas in Dalton, New York

The number of throws to get a million pounds would be at least 20 since 2^20 is 1,048,576.

This would require throwing at least 19 tails sequentially, which has a probability of 1/ 2^19 or 1/524,288, or approximately 0.000190735 percent.

The amount you should pay to play the game is less than or equal to the expected value of the winnings.

After the first turn the expected winnings are 1/2*2 = 1 pound.

After two turns the expected winnings are 1 + 1/4*4 = 2 pounds.

After n turns the expected value is n pounds; therefore, one would pay any amount of money, assuming that there's no Gambler's ruin.