What this calculator does
Lottery odds come from combinations. The order the numbers are drawn in does not matter, so the count of possible tickets is the number of ways to choose the drawn quantity from the pool, written as C(n, k). A separate bonus pool multiplies that count again.
The figures grow much faster than people expect as the pool size rises. Choosing 6 numbers from 45 gives 8,145,060 possible combinations; choosing 6 from 49 gives 13,983,816. Four extra balls in the pool nearly doubles the odds against you.
The formula
The number of possible tickets is C(n, k), which is n! divided by k!(n − k)!. If there is a second pool for a bonus ball, its own combination count is calculated the same way and the two are multiplied. The odds of any one ticket winning are 1 in that total.
| Term | Meaning |
|---|---|
| Combination | A selection where order does not matter, which is what a lottery draw is: the same six numbers in any sequence is the same ticket. |
| C(n, k) | The number of ways to choose k items from n, read as "n choose k". |
| Bonus pool | A separate set of balls drawn independently of the main numbers, which multiplies the total number of possible outcomes. |
The inputs explained
| Field | What to enter |
|---|---|
| Main numbers to choose from | How many numbers are in the main pool to choose from. |
| Main numbers drawn | How many main numbers are drawn. |
| Bonus pool size (0 for none) | The size of a separate bonus pool, if the game has one. Leave at zero if not. |
| Bonus numbers drawn | How many bonus numbers are drawn from that separate pool. |
| Tickets bought | How many different tickets are bought, used to show how the odds scale with volume. |
When to use it
Understanding a specific game
Different lotteries use different pool sizes, and the odds vary by orders of magnitude between them, so the format matters more than any strategy.
Seeing what more tickets buy
Ten tickets improve the odds by a factor of ten, which sounds substantial until it is applied to a figure in the millions.
Checking a published figure
Lottery operators publish the odds for their games, and recalculating them from the format confirms where the number came from.
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.
How do the odds change with the size of the number pool?
Six main numbers drawn from a range of pool sizes.
| Numbers in the pool | Odds of matching everything |
|---|---|
| 40 | 1 in 3,838,380 |
| 45 | 1 in 8,145,060 |
| 49 | 1 in 13,983,816 |
| 59 | 1 in 45,057,474 |
Questions
Do some number combinations come up more often?
No. Every combination is equally likely on every draw, and past results carry no information about future ones. Picking numbers others avoid does not change the chance of winning, though it can reduce the chance of sharing a prize.
Does buying more tickets help?
It scales the odds proportionally, so ten tickets are ten times better than one. Starting from odds of one in several million, that improvement is real arithmetic and still leaves a very small number.
What about the smaller prize divisions?
This calculator gives the odds of matching every drawn number. Partial matches are far more likely and have their own combination counts, which vary by game and prize structure.
Why does a bonus pool matter so much?
It multiplies rather than adds, since the bonus is drawn independently of the main numbers. A bonus ball drawn from a pool of 20 makes the top prize twenty times harder to win than the main numbers alone.
For the underlying combination arithmetic, see the permutations and combinations calculator. For odds and probability in general, see the probability calculator.