Lottery Odds Calculator

What this calculator does

See exactly how long the odds are in a pick-style lottery, where you choose a set of numbers. The draw picks its numbers, and the order doesn't matter. So the number of possible tickets is a combination, a count of the ways to choose a group. For example, it's how many ways there are to choose 5 numbers from 69, or 6 from 49.

Some games draw a separate bonus ball from its own pool. Every main combination can pair with every bonus number, which multiplies the count. Exactly one of all those tickets wins the jackpot.

This lottery odds calculator counts the combinations for any game you describe and gives the jackpot odds. It also lists the odds of every match tier, such as 4 of 5 plus the bonus. And it puts the jackpot in perspective with its expected value. That's what one ticket gets back on average from the jackpot, compared with what it costs. The answer is plain math, not a prediction, and no way of picking numbers changes it.

Where it's used

Why this math matters outside the classroom:

• Probability class. Students compute C(49, 6) = 13,983,816 to see why a 6-from-49 jackpot is a 1 in about 14 million chance.
• Expected value. Statistics students show that a $2 ticket with a $100 million jackpot at 1 in 292,201,338 returns about $0.34 on average.
• Comparing games. Players compare a 5-of-69 game with a 6-of-49 game and see the bonus ball makes the first about 21 times harder to win.
• Consumer education. Personal finance teachers use the break-even jackpot of $584 million to discuss why lottery tickets lose money on average.
• Smaller prizes. Matching exactly 3 of 6 in a 6-from-49 game has 246,820 winning tickets, odds of about 1 in 56.7.

Grade level and standards: Grades 11–12; Common Core State Standards for Mathematics HSS.CP.B.9, HSS.MD.A.2. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

1. Enter how many main numbers there are to choose from and how many are drawn.
2. Enter the size of the bonus ball pool, or 0 if the game has none.
3. Enter the advertised jackpot and the ticket price for the expected value.
4. Read the jackpot odds, then open the table for the odds of each match tier.

What the inputs mean

Main numbers to choose from

For a 6-from-49 game, enter 49.

Bonus ball pool

This is for games that draw the bonus from a separate pool. Some games draw an extra ball from the same main pool. Those follow a different calculation and aren't modeled.

Advertised jackpot

The advertised jackpot is usually the total of yearly payments spread over many years. A one-time lump sum is smaller, and taxes reduce it further. Winners may also share it. So the expected value shown here is generous.

Formula and method

main combinations = C(N, k) = N! ÷ (k! × (N − k)!)
tickets = C(N, k) × B   (B = bonus pool size, or 1 with no bonus)
jackpot odds = 1 in tickets
ways to match m main numbers = C(k, m) × C(N − k, k − m), times 1 with the bonus or B − 1 without it
expected value per ticket = jackpot ÷ tickets − price

N is how many main numbers there are and k is how many are drawn. B is the size of the bonus pool, and m is how many main numbers a ticket matches. C(N, k) counts the ways to choose k numbers from N when order doesn't matter. The ! sign (factorial) means multiply all the whole numbers down to 1, so 5! = 5 × 4 × 3 × 2 × 1.

To match exactly m of the k winning numbers, a ticket needs m of the winners and k − m of the N − k losing numbers. Multiplying those two counts gives the number of such tickets. For the bonus, one bonus number matches and B − 1 don't.

Expected value is each outcome's payout times its chance, all added up. Counting the jackpot alone shows the most it can add. Real games also pay smaller prizes. But tax, shared jackpots and the smaller lump sum pull the figure down.

Worked example

A game draws 5 numbers from 69 and a bonus ball from 26. The jackpot is advertised at $100,000,000 and a ticket costs $2.

1. C(69, 5) = 11,238,513 ways to choose the main numbers.
2. 11,238,513 × 26 = 292,201,338 different tickets.
3. Jackpot odds: 1 in 292,201,338.
4. Expected jackpot value: $100,000,000 ÷ 292,201,338 = $0.34 per ticket.
5. $0.34 − $2.00 = −$1.66 per ticket from the jackpot alone.

Result: The jackpot odds are 1 in 292,201,338. On average, a $2 ticket gets back about 34 cents from the jackpot, an expected loss of about $1.66.

Understanding the result

Odds of 1 in 292 million are hard to picture. Buying one ticket a week, you'd expect to wait several million years for a jackpot. Buying 10 tickets makes the odds 1 in about 29 million. That's ten times better, but still remote.

The break-even jackpot is the size at which the jackpot alone would pay back the ticket on average. Even above it, taxes and shared wins usually keep the real expected value negative. Lotteries are designed to pay out less than they take in.

Assumptions and limitations

• Every combination is equally likely, and each ticket has different main numbers.
• One bonus number is drawn from a separate pool, and the jackpot needs all main numbers plus the bonus.
• The expected value counts one jackpot winner and ignores smaller prizes and taxes.
• Prize amounts for lower tiers vary by game, so only their odds are shown.
• Games with two bonus balls, a bonus drawn from the main pool, or multipliers aren't modeled.
• It handles up to 100 main numbers and 10 drawn.

Frequently asked questions

How this calculator is checked

Method: Combinatorics: C(N, k) tickets times the bonus pool; tier counts C(k, m) × C(N − k, k − m); expected value from the jackpot alone. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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