Lottery EV

Hi Wizard/regs,

I've read your lottery FAQ and noticed you advocate using Poisson to determine the EV when splitting the jackpot. My question regards the non-jackpot division prizes in a lottery where NONE of the prizes are fixed.

If prizes for each division are split among expected winners, is the Poisson method applicable to every division? (ie is this the mathematically correct way of looking at it?)

thanks!

In a simple Lotto game, usually the bottom tier prize is a fixed payment. Others above can be a pari-mutual distribution based upon pool contribution and numbers of winners. The pool contribution is constant, the winners vary. This is a mixed payout.

The EV for an unknown number of winners can be independantly calculated knowing all the facts about the game itself. The expected number of winners and expected payment in a minor prize tier is discovered in the following example;

Lotto 6/40 Price $1 Odds to win Jackpot is 40C6 (combinations of 6 numbers from a field of 40) equals 1 in 3838380
The operator takes $0.40 per ticket. This example presumes that all 3838380 combinations are purchased without duplicates.

Jackpot is 6 from 40 balance of $0.60 minus all payouts
2nd Prize is 5 of 6 from 40 10% contribution
3rd prize is 4 of 6 from 40 10% contribution
4th prize is 3 of 6 from 40 and pays $3 (fixed prize)

2nd Prize: 40C6 / (6C5)*(34C1) The denominator is the expected number of winners, solving yields the "for" odds.
In this case the denominator solves as 6*34 = 204 expected winners, the Odds are 1 in 18815.6. Since 10% is placed into the pool, the 204 expected winners receive $1881.56. If there are fewer than normal winners, the prize increases.

Likewise 3rd Prize is calculated as 40C6 / (6C4*34C2) = 8415 expected winners with 10% contribution pays $45.61

4th prize is calculated as 40C6 / (6C3*34C3) = 119680 expected winners of $3. The odds are 1 in 32.07 the fixed contribution equals 3/32.07 = 9.354%

The expected contribution is 29.354 cents. The total pool is 60 cents. The expected Jackpot contribution is 60-29.354 = 30.646 cents for an expected jackpot of 0.30646 * 3838380 = $1176309.94.

Its complex in the real world, certain elements remain fixed, and other vary. Even knowing these parameters, a database of actual winners per tickets sold must be consulted. Re: Is the game hitting certain payouts more, or less frequently than average over the limited history? based upon historical data one can surmise how many winners of a certain prize-tier is expected DEPENDENT UPON HISTORY Vs. MATHEMATICAL INTERPRETATIONS.