A simple coin game is as follows: we have a box, which starts empty. P1 flips a fair coin. If it comes up heads, P2 gets the contents of the box, and the game ends. If it comes up tails, P1 puts a dollar in the box and they flip again; this repeats until it comes up heads. a. With what probability will P2 win exactly 10 units?
b. Write ğ[infinity] = Ã[infinity] ğ=0 ğ ğ . Show that (1 â ğ)ğ[infinity] = 1, so that ğ[infinity] = 1 1 â ğ .
c. Show that âï¸[infinity] ğ=0 ğğğ = âï¸[infinity] ğ=0 ğ ğ ! + ğ âï¸[infinity] ğ=0 ğ ğ ! + ğ 2 âï¸[infinity] ğ=0 ğ ğ ! + · · · (look carefully at the limits of the sums!) and so show that âï¸[infinity] ğ=0 ğğğ = ğ (1 â ğ) 2 .
d. What is the expected value of the game? (you may find the results of the two previous subexercises helpful; theyâre not there just for show).
e. How much should P2 pay to play, to make the game fair?