There is a one hole golf competition played by 1000 people, entry is 1 dollar.
Contestants play a par 3 once, and if they get a hole-in-one, they win. If multiple people get a hole-in-one, the jackpot is shared between them. If no one gets a hole in one, the house takes the money.
Everyone has an equal chance of getting a hole-in-one,$\dfrac{1}{100}$.
The question is, what are the expected returns for a player in this competition?
I tried this using an exhaustive tree diagram/spreadsheet, but it got very tedious.Anyone know a nice way to do this?
The chance that nobody will hit a hole in one is $$ \left( \frac{99}{100} \right)^{1000} \approx 0.000043 $$ This makes the total expected house take $0.043$.
So since everybody will have an equal expectations, the expected return for any given player is $$ 1- \left( \frac{99}{100} \right)^{1000} \approx 0.999957 $$