What is the formula for the expected duration of a 15 sided dice game?

Question

There are n participants. Each participant has a 15 sided dice. What is the formula for the expected number of rolls required for a 1 to be rolled in the game?

Answer 1

Let p = P( get it on first round ) = 1 - (14/15)^n

E( number of rounds ) = P( get it on first round) + P( dont get it on first round) ( 1 + E(number of rounds) )

= p + (1-p) (1 + E( number of rounds ))

Let e = E( number of rounds ). Then we have:

e = p + (1-p)*(1+e)

(1-(1-p))e = p + 1 - p

e = 1/p

Which makes sense because when p=1, the expected number of tries is 1, and when p goes to 0, the expected number of rounds goes to inf.