The mathematics of odds
A pari-mutuel prize is a pool divided by winners, which makes your payout depend on other people's choices. We derive the co-winner distribution and show what popular number patterns cost.
There are two ways a lottery can decide what you win. A fixed prize is a promise: match four numbers on UK Lotto and you receive £50, whoever else does. A pari-mutuel prize is a division: a pool of money is set aside for the division, and everyone who qualifies shares it equally. Nearly every jackpot in the world is pari-mutuel, which means the number on the billboard is not what you win — it is what the winners collectively win.
Take UK Lotto, whose Game Procedures state the arithmetic openly:
So UK Lotto is a hybrid: one pari-mutuel division on top, five fixed ones underneath. Work a draw with £40,000,000 of sales:
| Step | Calculation | Result |
|---|---|---|
| Prize fund | 50% × £40,000,000 | £20,000,000 |
| Reserve fund held back | 9.34% × £20,000,000 | £1,868,000 |
| Jackpot allocation | 8.88% × £40,000,000 | £3,552,000 |
| Fixed prizes (Match 2–5+Bonus) | from the remainder | pays out as promised |
If one ticket matches six, that ticket takes £3,552,000 — except Allwyn undertakes to "top up the Jackpot so that each Winning Lotto Entry in the Jackpot Prize category receives at least £1million", so a floor applies. If four tickets match six, each receives £3,552,000 ÷ 4 = £888,000, topped up to £1,000,000 each under the same rule. Same draw, same numbers, quarter of the money — because three strangers picked the same six.
Australia runs the purer form — every division is a percentage of the pool, none is fixed. The NSW/ACT Lotto Game Rules set the Prize Pool at "not less than fifty five percent (55%) of Subscriptions", and The Lott publishes the division split of that pool. For Saturday Lotto, effective from Draw 4575:
| Division | Share of the prize pool |
|---|---|
| Division 1 | 34.10% |
| Division 2 | 3.27% |
| Division 3 | 4.95% |
| Division 4 | 7.35% |
| Division 5 | 11.50% |
| Division 6 | 38.83% |
| Total | 100.00% |
Work a draw with A$20,000,000 of subscriptions:
Prize pool = 55% × A$20,000,000 = A$11,000,000 Division 1 pool = 34.10% × A$11,000,000 = A$3,751,000 With 4 winning entries: A$3,751,000 ÷ 4 = A$937,750 each
The dividing rule is stated in the legislation rather than left to practice. Queensland's Lotteries Rule says "the prize pool for a division of a drawing must be divided in equal shares among the winners in the division", and New Zealand's Lotto Rules 2025 that "the prize money allocated to that division must be shared equally between each player or other claimant with a winning selection in the division."
Notice Division 6 takes a larger share of the pool than Division 1 does. That is not an error: Division 6 has hundreds of thousands of winners per draw and Division 1 usually has none. Sharing is not an edge case in a pari-mutuel game; it is the normal state of every division except the top one.
Let S be the number of lines sold and p the probability that one line matches. If people picked at random, the number of winning lines follows a Binomial(S, p) distribution, and because p is minuscule and S is huge, that is essentially a Poisson distribution with mean:
λ = S × p
P(exactly k winners) = e^(−λ) × λ^k / k!
The number you actually care about is your share of the pool given that you have won. If X is the number of other winners, your share is 1/(1 + X), and its expectation has a compact closed form:
E[1/(1+X)] = Σ (k ≥ 0) e^(−λ) λ^k / (k! (k+1)) = (1/λ) Σ (k ≥ 0) e^(−λ) λ^(k+1) / (k+1)! = (1 − e^(−λ)) / λ
That single expression — (1 − e^(−λ)) / λ — is the whole economics of jackpot sharing. Some values:
| λ (expected winners) | P(no winner at all) | Your expected share of the pool |
|---|---|---|
| 0.10 | 90.5% | 95.2% |
| 0.25 | 77.9% | 88.5% |
| 0.50 | 60.7% | 78.7% |
| 1.00 | 36.8% | 63.2% |
| 2.00 | 13.5% | 43.2% |
| 3.00 | 5.0% | 31.7% |
| 5.00 | 0.7% | 19.9% |
Applied to US Powerball, where p = 1/292,201,338 and a $2 ticket buys one line:
| Lines sold | λ | Your expected share | Effective jackpot on a $500m advertised prize |
|---|---|---|---|
| 50,000,000 | 0.171 | 91.9% | $459.6m |
| 100,000,000 | 0.342 | 84.7% | $423.4m |
| 200,000,000 | 0.685 | 72.4% | $362.1m |
| 300,000,000 | 1.027 | 62.5% | $312.6m |
| 400,000,000 | 1.369 | 54.5% | $272.4m |
This is the reason a rolling jackpot does not improve as fast as the headline suggests. Rollovers attract sales; sales raise λ; a rising λ shrinks the share. The two effects fight, and past a certain point the sharing effect wins. It is the single biggest correction to naive expected-value reasoning — our EV calculator lets you switch it on and off.
Everything above assumes tickets are spread uniformly over the combination space. They are not, and the departure is enormous.
The 14 January 1995 UK draw. The ninth Lotto draw produced 133 jackpot winners, each receiving £122,510 from a jackpot pool of 133 × £122,510 = £16,293,830, against a total prize fund of £41,415,050 (full result). The game was 6-from-49, so p = 1/13,983,816.
How impossible is 133? We do not even need the sales figure. Sales must exceed the prize fund, so at £1 a line the absolute ceiling on λ is 41,415,050 / 13,983,816 = 2.96 — and that assumes the physically impossible case of 100% of sales returned as prizes. At λ = 2.96, the probability of seeing 133 or more winners is about 10^−165. With the real prize-fund share it is smaller still. The conclusion is not statistical, it is arithmetical: those numbers were not chosen randomly. They form a pattern on the play slip, and thousands of people marked it.
Powerball, 30 March 2005. The drawing produced a record 110 second-tier winners — a tier whose odds were then roughly one in three million. The Multi-State Lottery Association's executive director said it was "out of the realm of possibility" and opened a fraud investigation. The explanation was a fortune-cookie slip printed with 22, 28, 32, 33, 39 — five of the six drawn numbers (account). Eighty-nine winners took $100,000 and twenty-one who had bought Power Play took $500,000. No cheating; just correlated choices.
The mechanisms are well known and all point the same way:
None of this changes your probability of winning by a single digit. It changes what winning is worth. That is why a random machine pick is defensible on strictly financial grounds even though it cannot improve your odds — the argument is laid out in quick picks vs chosen numbers, and you can score your own line for crowding with the number-sharing risk tool.
Your odds are set by the matrix and nothing can move them. Your payout is set by the pool and by how many other people made the same choice you did — and that, unlike the odds, is something a player can influence.
Last verified: 2026-08-29