What this calculator tells you
Many lotteries ask players to choose K numbers from a pool of N, and the jackpot goes to anyone whose K numbers all match the draw. For that format, the calculator turns the two numbers you enter into the exact jackpot odds, expressed as "1 in some number" and as a percentage. If you also enter a number of tickets, it shows the probability that at least one of them hits the jackpot. The default values, 6 numbers out of 45, are used throughout this page as the running example.
It is a probability tool, not a prediction tool. It cannot tell you which numbers to pick, because every combination is equally likely, and it does not recommend buying tickets. Typical uses are sanity-checking a claim you saw somewhere, comparing the difficulty of different formats, or checking combinatorics homework.
The math: combinations
If the order of the numbers does not matter, the number of ways to choose K numbers from N is the binomial coefficient C(N, K) = N! Γ· (K! Γ (N β K)!). For a 6-of-45 game, C(45, 6) = (45 Γ 44 Γ 43 Γ 42 Γ 41 Γ 40) Γ· (6 Γ 5 Γ 4 Γ 3 Γ 2 Γ 1) = 8,145,060. Only one of those combinations is the winning one, so a single ticket has a jackpot chance of 1 in 8,145,060, or about 0.0000123%.
The calculator multiplies and divides step by step instead of building huge factorials, and it keeps the intermediate values as exact integers (BigInt), so the result stays exact even when the pool is large. Because C(N, K) = C(N, N β K), it always uses the smaller of the two, which keeps the loop short.
For multiple tickets, assume each ticket is an independent chance with jackpot probability p = 1 Γ· C(N, K). The chance of at least one jackpot in m tickets is 1 β (1 β p)m. When p is tiny, this is almost exactly m Γ p, which is why buying twice as many tickets roughly doubles a very small chance, and never gets near certainty. The page computes it with a numerically stable formula so very small probabilities do not collapse to zero.
Prize tiers in a 6/45 draw with one bonus ball
Many lotteries add lower prize tiers for partial matches, and some use a bonus ball. As an illustration, suppose the draw takes 6 main numbers and then one bonus ball from the remaining 39 numbers. The table below counts the tickets in each tier. It was computed from the combination formula when the page was built. These are counts and probabilities only; no prize amounts are implied, and real games set their own tiers and rules.
| Tier | Winning tickets | 1 in | Probability |
|---|---|---|---|
| All 6 main numbers | 1 | 8,145,060 | 0.0000123% |
| 5 main numbers + the bonus ball | 6 | 1,357,510 | 0.0000737% |
| 5 main numbers, no bonus ball | 228 | 35,724 | 0.0028% |
| 4 main numbers | 11,115 | 733 | 0.1365% |
| 3 main numbers | 182,780 | 45 | 2.2441% |
The five rows together cover 194,130 of the 8,145,060 possible tickets, about 2.38%, or roughly 1 ticket in 42.0. Almost all of that comes from the 3-number tier. The counting logic: for exactly j matches among your 6 numbers, choose j from the 6 winning numbers and the other 6 β j from the 39 non-winning numbers, giving C(6, j) Γ C(39, 6 β j). The bonus-ball tiers split the 5-match case: one of the 39 other numbers is the bonus ball, so 1 of 39 is "with bonus" and 38 are "without".
How the jackpot odds change with the format
The table lists a few single-pool formats, so you can feel how quickly the number of combinations grows. These are illustrations of the formula, not descriptions of any specific lottery.
| Format | Formula | Possible combinations |
|---|---|---|
| 6 of 45 | C(45, 6) | 8,145,060 |
| 6 of 49 | C(49, 6) | 13,983,816 |
| 5 of 35 | C(35, 5) | 324,632 |
| 6 of 59 | C(59, 6) | 45,057,474 |
| 5 of 90 | C(90, 5) | 43,949,268 |
Adding even a few numbers to the pool, or one more number to match, changes the odds by a large factor. Going from 6-of-45 to 6-of-49 makes the jackpot roughly 1.7 times harder. Games that draw from two separate pools, for example 5 numbers from one pool and 1 from another, multiply the combinations for each pool. This calculator handles one pool, so for a two-pool game compute each pool separately and multiply the results by hand.
Worked examples (verified with code)
Example 1: one ticket per week for a year. In 6/45 that is 52 tickets. The chance of at least one jackpot is 1 β (1 β 1/8,145,060)52 β 0.000638%, about 1 in 156,636.
Example 2: five years of weekly tickets. That is 260 tickets, giving about 0.003192%, or roughly 1 in 31,328. After five years the chance has gone from vanishing to still vanishing.
Example 3: a lot of tickets. Even 100,000 random independent tickets gives only about 1.22%. Buying as many random tickets as there are combinations (8,145,060) does not give certainty: the chance is about 63.2%, which is 1 β 1/e. Only a set of all distinct combinations guarantees the jackpot.
| Independent tickets (6/45) | Chance of at least one jackpot |
|---|---|
| 1 | 0.000012% |
| 52 | 0.000638% |
| 260 | 0.003192% |
| 1,000 | 0.01% |
| 100,000 | 1.22% |
| 8,145,060 | 63.21% |
Common misconceptions
- "Hot" numbers are more likely. Each draw is independent. A number that came up often last month has the same chance next time.
- "Overdue" numbers will come up soon. This is the gambler's fallacy. A coin that lands heads ten times in a row still lands heads half the time on the next flip.
- 1-2-3-4-5-6 never wins. It is exactly as likely as any other single combination. What differs is how many people choose popular patterns, which can mean a shared prize if it does come up.
- Spreading numbers or using patterns improves your odds. Only the number of distinct tickets matters, not their pattern.
- Quick-pick and hand-picked tickets have different odds. They do not. The draw is blind to how you chose.
Birthday-based picks illustrate the one practical difference. If you choose only numbers 1 to 31, you are limited to C(31, 6) = 736,281 combinations, about 9.0% of the 8,145,060 total. This does not change your chance of winning, but it concentrates many players on the same tickets.
Limits and when not to use it
- Games where order or position matters, such as picking a digit for each position, are counted with permutations or powers (for example 10 raised to the number of digits), not combinations.
- Games with two pools or bonus numbers cannot be done in a single run of this tool. Use the 6/45 table above as a pattern and multiply the combinations for each pool.
- Winning odds are not expected value. Lotteries return only part of their sales as prizes, so the average return per ticket is below the ticket price. In other words, the expected value of a ticket is negative.
- N above 500 is rejected, because the result would be too large to display meaningfully.
How to use the result
If you play at all, treat the cost like any other entertainment spending and set a firm limit in advance. The figures here show why no purchasing strategy turns a lottery into an investment: more tickets move a very small number slightly less small. Use the result to set realistic expectations, not to justify spending more. If gambling is causing you trouble, local support organizations can help.
Related tools: the lotto number generator for random picks, the random number generator, the percentage calculator, and the coin and dice roller for hands-on probability practice.
Frequently Asked Questions
Does the headline figure include the bonus ball?
No. The headline result is the chance of matching all K numbers out of N. Lower tiers that depend on a bonus number are not computed by the form itself. The 6/45 table above shows how to count them.
Can I use it for any country's lottery?
For any game where you pick K from a single pool of N and the order does not matter, yes. Games with two pools need each pool computed separately and multiplied. Always check the operator's current rules, since formats change.
What does entering a number of tickets do?
It shows the chance that at least one ticket hits the jackpot, 1 β (1 β p)tickets. It assumes each ticket is independent. For small p the result is close to tickets Γ p and slightly lower.
Why is every combination equally likely?
A fair draw selects numbers uniformly, so each of the C(N, K) combinations has probability 1 Γ· C(N, K). How you choose a ticket does not change its probability.
Why is the odds figure sometimes shown as an approximation?
When the number of combinations exceeds 1015, the page shows scientific notation instead of every digit. The calculation itself still uses exact integers.