Question

A player is given the choice to play this game. The player flips a coin until they get the first Heads. Points are awarded based on how many flips it took:

1 flip (the very first flip is Heads): 2 points
2 flips (the second flip was the first Heads): 4 points
3 flips (the third flip was the first Heads): 8 points
4 flips (the fourth flip was the first Heads): 16 points

and so on. If the player makes n flips, they win 2 n points

Now at most 32 flips can be made. If the first 32 flips are all Tails, then the player stops flipping (even though they didn't get a Heads yet) and wins 2 32 points. How many points on average does the player win? How much should the player be willing to spend to play this game?

Answer 1

Let X be the amount of winning points and K be the number of flips to get the first head. The probability distribution of X is,

for k = 1, 2, ..., 31

(Probability to get all tails in 31 flips)

Average number of points the player wins =

= 31 + 2

= 33

To have a positive return from the game, the player would be willing to spend at most $33 to play this game.