Expected Value of Lottery Jackpot, Given You Are A Winner

Reply #6

cottoneyedjoe — Wed, 20 Jul 2022 19:07:31 GMT

You're right. Those were numbers I obtained running simulations. I obviously didn't run enough since the frequencies are off, lol. After gave it some thought I realized the exact distribution really is just binomial and not something else, I couldn't edit the post and then I forgot and moved on to something else. Sorry

Reply #5

Orange71 — Wed, 20 Jul 2022 19:05:22 GMT

N and n are the same. Sorry about that sloppiness. Trying to fight this awful editor was an uphill battle.

Reply #4

Orange71 — Wed, 20 Jul 2022 18:16:41 GMT

Actually, your math doesn't seem to be quite right on the first example. You set N = 1 / p, where N = 575757. OK, no problem. However, let's expand the terms for k = 0 and k = 1 in my equation, the cases where there is no other winner (i.e. besides you/me) and one other winner.

P(k=0) = [(n-1)C(0)] * (1-p)^(N-1) * p^0 = (1) * [(1-p)^(N-1)] = (1-p)^(N-1)

P(k=1) = [(n-1)C(1)] * [(1-p)^(N-2)] * p = (n-1) * [(1-p)^(N-2)] * p = (p*n - p) * [(1-p)^(N-2)] = (1-p) * [(1-p)^(N-2)] = (1-p)^(N-1... [ More ]

Reply #3

cottoneyedjoe — Wed, 20 Jul 2022 03:53:43 GMT

You're right, I see it now. Poisson is also the wrong one because it has infinite support where as this distribution is finite. Thanks for the book recommendation, I got a shelf full of Dover editions. I can't believe more college courses don't use them just on the price alone, but they are written without much filler and leave the details to the reader, probably not suitable for most college students except math majors and grad students.

Reply #2

Orange71 — Tue, 19 Jul 2022 23:56:48 GMT

The underlying distribution is Binomial. Poisson is an approximation of the Binomial Distribution for cases when p is small, and N is large (both of which should be reasonable for just about any lottery).

By the way, if you want a condensed probability and statistics primer, I recommend Principles of Statistics by M.G. Bulmer, who was a professor at Cambridge University in the 1960s. It's a Dover Edition, $15. I think the latest edition was published in 1967. It was published in an era w... [ More ]

Reply #1

cottoneyedjoe — Tue, 19 Jul 2022 16:59:48 GMT

This is an interesting topic, nice write up. In the past I have run simulations to find the probability distribution of the number of jackpot winners (under the assumption every ticket is a quick pick to simplify things) varying the parameters of total number of combinations and number of tickets sold.

For example, in a 5/39 game, which is common in many states, there are 575757 combinations (a more manageable number). If 575757 quick picks are sold, the likelihoods of 0, 1, 2, 3, and 4+ jac... [ More ]

Expected Value of Lottery Jackpot, Given You Are A Winner

Orange71 — Mon, 18 Jul 2022 21:07:38 GMT

When you play a lottery and win a pari-mutuel Jackpot, there's always the possibility that one or more other players will also have won, and you'll be sharing (splitting) the prize money. Can you calculate what the Expected Value is of the Jackpot prize, given you are a winner? Certainly!

We need to assume something first: every ticket bought is a Quickpick - so any given set of numbers (a single ticket) can be randomly replicated on any other ticket. Suppose there are N such tickets sold (1... [ More ]