Golf competition

Question

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?

$\sum_{n = 1}^{N} \frac{N}{n} \left({N \atop n}\right) p^n (1 - p)^{N - n}$ where $p = 1/100$ and $N = 1000$ ? — user66081, May 12 '17 at 22:48
Sorry, I had to think about that for a while. Thanks so much, very insightful — , May 12 '17 at 23:01

score 1 · Accepted Answer · answered May 12 '17 at 23:06

1

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 $$ 1- \left( \frac{99}{100} \right)^{1000} \approx 0.999957 $$

answered May 12 '17 at 23:06

Mark Fischler

41,743

Less the entry fee. $E[X] = -(\frac {99}{100})^{999}$ – Doug M May 12 '17 at 23:15

Golf competition

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