Lottery Odds Calculator
The odds of matching a lottery draw — jackpot and each lower division — for any pick-r-from-n game, with or without a supplementary ball.
A draw of r numbers from n has C(n, r) equally likely outcomes, so one ticket has one chance in that.
How the lottery odds calculator works
A draw of r numbers from n has C(n, r) equally likely outcomes, so one ticket has one chance in that. Matching exactly m of the drawn numbers means choosing m from the r drawn and r − m from the n − r not drawn: C(r, m) × C(n − r, r − m) ÷ C(n, r).
A supplementary or bonus ball drawn from a separate pool multiplies the jackpot odds by that pool size.
Formula: P(match m) = C(r, m) C(n − r, r − m) / C(n, r)
Worked examples
| Inputs | Chance of matching exactly m | Note |
|---|---|---|
| Australian Saturday Lotto: 6 from 45 | 0.00001228% | 1 in 8,145,060 |
| Matching 4 of 6 | 0.13646308% | about 1 in 733 |
| UK Lotto: 6 from 59 | 0.00000222% | 1 in 45,057,474 |
FAQFrequently asked questions
What are the odds of winning the lottery?
For a 6-from-45 game, 1 in 8,145,060 per ticket. For UK Lotto (6 from 59), 1 in 45,057,474.
Do more tickets help?
Proportionally, yes — and negligibly. Ten tickets take you from 1 in 8.1 million to 1 in 815,000, still far below the chance of being struck by lightning this year.
Are some numbers luckier?
No. Every combination is equally likely. Picking unpopular numbers does not raise your chance of winning, only your share if you do.
Why are the lower divisions so much easier?
Because many combinations match partially. Matching 3 of 6 in a 45-ball game happens about 1 time in 37.
Is a syndicate better value?
It raises your chance in exact proportion to tickets held, and cuts the prize the same way. The expected value is unchanged; only the variance falls.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Australian Curriculum (ACARA) — Mathematics — the terms and methods taught in Australian schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.