Lottery Math Toolkit
Combinatorial wheels · unpopularity-weighted picks · EV analysis for US multi-state lotteries
Why unpopular numbers help
Your probability of winning any prize tier is fixed by the game's matrix and is identical for every ticket. But if you do hit the jackpot, it's split equally among all jackpot winners. Actual ticket purchases heavily over-weight dates (1-31), cultural lucky numbers, and patterns like diagonals on the play slip. Picking numbers players avoid raises E[payout | win] — real EV improvement, no probability handwaving.
Cook & Clotfelter (1993); Chernoff (1999); Simon (1998)
What wheeling actually does
A covering design C(v, k, t, g) is a set of k-sized tickets drawn from v candidates such that if at least t of the v candidates are in the drawn set, at least one ticket matches g of them. Example: our 6-candidate 4-ticket Powerball wheel guarantees a 3-match if any 3 of your 6 picks are drawn. Wheels cost more tickets but give certainty over a conditional outcome.
Bluskov, La Jolla Covering Repository
When is a ticket +EV?
Let J be the advertised jackpot, c the cash ratio, τ tax rate, and π(J) the probability no one else wins. Expected payout per ticket ≈ (J·c·(1−τ)·π(J))/odds + Σ fixed-tier contributions. When this exceeds the ticket price, it's mathematically +EV. This does happen, briefly, for Powerball and Mega Millions at very high rollover jackpots — but variance is still extreme. The Kelly bet size for any realistic bankroll is effectively zero.
Thorp (1984); also any intro martingale text