What this calculator does
Lottery odds come down to a single combinatorics question: out of all the possible ways to choose a set of numbers from a pool, how many of them match the one winning combination. This calculator applies that formula directly to whatever lottery format you enter, rather than relying on a headline figure quoted for a specific game.
The maths is the standard "combinations" calculation: choosing 6 numbers from a pool of 45, for example, has millions of equally likely outcomes, and only one of them is the winning draw. Adding a separate bonus-ball pool, as many lotteries do, multiplies the odds against you again.
The formula
The number of ways to choose the drawn numbers from the pool is calculated using the combinations formula, C(pool size, numbers drawn), which counts unordered selections without repetition. The odds of a full match are 1 in that number. If a bonus ball is drawn from a separate pool, the combined odds multiply the main combinations by the bonus-ball pool size.
| Term | Meaning |
|---|---|
| Combination | A selection of numbers where order does not matter, matching how lottery draws are checked. |
| C(n, r) | The number of ways to choose r items from a pool of n, read as "n choose r". |
| Bonus ball | An extra number drawn from a separate pool, used by some lotteries to add another prize tier or reduce the jackpot odds further. |
The inputs explained
| Field | What to enter |
|---|---|
| Total number pool | The total size of the main number pool the draw is made from. |
| Numbers drawn | How many numbers are drawn from that pool to form a winning combination. |
| Extra bonus-ball pool (0 if none) | The size of a separate bonus-ball pool, if the lottery draws an extra number from a different range. Leave at 0 if there is no bonus ball. |
When to use it
Checking a specific lottery’s odds
Entering a game’s actual pool size and draw count gives its real odds of a full match, useful for comparing against the odds a lottery operator advertises.
Comparing different lottery formats
Two lotteries with similar-sounding rules can have wildly different odds once the pool size and draw count are worked through properly, which this makes visible side by side.
Understanding what a bonus ball does to the odds
Adding a bonus-ball pool shows exactly how much steeper the odds of a full jackpot match become compared with the main draw alone.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
Odds of a full match as the number pool grows
How much the odds lengthen as the pool of possible numbers grows, with the count drawn held at 6.
| Number pool | Ways to draw 6 from the pool | Odds of matching all 6 numbers |
|---|---|---|
| 35 | 1,623,160 | 1 in 1,623,160 |
| 40 | 3,838,380 | 1 in 3,838,380 |
| 45 | 8,145,060 | 1 in 8,145,060 |
| 49 | 13,983,816 | 1 in 13,983,816 |
| 50 | 15,890,700 | 1 in 15,890,700 |
| 59 | 45,057,474 | 1 in 45,057,474 |
Questions
Why are lottery odds so much longer than they seem?
Because the formula counts every possible unordered combination of numbers, and that count grows extremely fast as the pool size increases, far faster than most people expect from a simple increase in the range of numbers.
Does buying more tickets improve my odds proportionally?
Yes, in the sense that two different tickets double your chances of one of them matching compared with one ticket, but the base odds for any single ticket remain those calculated here.
Is this the same as the general combinations calculator?
It uses the same underlying nCr formula as a general permutations and combinations calculator, but is set up specifically for the lottery framing of pool size, numbers drawn and an optional bonus ball, with the odds already worked out for you.
Does the order the numbers are drawn in matter?
No. Lottery draws are checked as an unordered set, which is exactly what the combinations formula assumes, unlike a permutation where order would matter.
For the underlying maths applied more generally, see a permutations and combinations calculator. For related odds questions in card games, a poker probability calculator uses the same nCr formula.