How Are Lottery Odds Calculated? Combinations Explained Step by Step

By KT Apps · Updated October 16, 2026

About this guide

This article is about mathematics, not about winning. It does not recommend buying lottery tickets, and nothing here can improve your chances. Lottery Simulator is a simulation for learning about probability: it does not sell tickets, take payments or pay prizes. If gambling is causing you problems, see the help section at the end.

The one formula you need: combinations

In most lotto-style games you choose a set of numbers, and the order does not matter: 3, 12, 25 is the same ticket as 25, 3, 12. Counting unordered selections is exactly what the combinations formula does. The number of ways to choose r numbers from a pool of n is written C(n, r), or "n choose r":

C(n, r) = n! ÷ (r! × (n − r)!)

The exclamation mark means factorial: 5! = 5 × 4 × 3 × 2 × 1 = 120. In practice you never need the huge full factorials, because most of them cancel. C(n, r) is simply the first r numbers counting down from n, multiplied together, divided by r!.

If every possible ticket is equally likely to be drawn, and only one ticket matches all the numbers, the jackpot probability is 1 ÷ C(n, r). That is the whole idea.

Worked examples, step by step

Example 1: choose 6 numbers from 49

1. Multiply the 6 numbers counting down from 49: 49 × 48 × 47 × 46 × 45 × 44 = 10,068,347,520.
2. Work out 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720.
3. Divide: 10,068,347,520 ÷ 720 = 13,983,816.
4. So there are 13,983,816 different tickets, and one ticket has a 1 in 13,983,816 chance of matching all six.

Example 2: choose 7 numbers from 37

1. Multiply 37 × 36 × 35 × 34 × 33 × 32 × 31.
2. Divide by 7! = 5,040.
3. The result is 10,295,472 possible tickets, so the chance of matching all seven with one ticket is 1 in 10,295,472.

In a spreadsheet the same calculation is one function: =COMBIN(49,6) returns 13,983,816 and =COMBIN(37,7) returns 10,295,472, in both Excel and Google Sheets.

Odds of matching only some numbers

Smaller prizes usually go to tickets that match some, but not all, of the drawn numbers. To count tickets that match exactly m of the r winning numbers, choose m numbers from the winners and the rest from the losers, then multiply:

Tickets matching exactly m = C(r, m) × C(n − r, r − m)

Divide by the total number of tickets to get the probability. Statisticians call this the hypergeometric distribution: drawing without replacement from a pool that contains a fixed number of "good" numbers. For a 6-from-49 draw, for example, matching exactly 3 counts C(6, 3) × C(43, 3) = 20 × 12,341 = 246,820 tickets out of 13,983,816, about 1 in 57.

This is why small wins feel common while the jackpot almost never arrives: there are hundreds of thousands of ways to match a few numbers, and exactly one way to match them all.

Odds versus probability

People often use the words interchangeably. Strictly, a probability of 1 in 4 corresponds to odds of 3 to 1 against, because there are 3 losing outcomes for every winning one. Lottery operators usually print "1 in X", which is the probability.

Bonus numbers from the same pool: Loto 7's six prize tiers

Japan's Loto 7 uses the 7-from-37 format from Example 2, with one twist. Each draw picks 9 different numbers from 1 to 37: 7 main numbers and then 2 bonus numbers. Your ticket is still 7 numbers, and the bonus numbers only matter for the 2nd and 6th prizes.

To count tickets for each tier, split the 37 numbers into three groups: the 7 main numbers, the 2 bonus numbers and the 28 numbers that were not drawn. A ticket that matches m main numbers and b bonus numbers can be built in C(7, m) × C(2, b) × C(28, 7 − m − b) ways. It is the same hypergeometric idea as above, with three groups instead of two.

• 1st prize, all 7 main numbers: 1 ticket, 1 in 10,295,472
• 2nd prize, 6 main numbers plus a bonus number (6+B): 7 × 2 = 14 tickets, about 1 in 735,391
• 3rd prize, 6 main numbers without a bonus: 7 × 28 = 196 tickets, about 1 in 52,528
• 4th prize, 5 main numbers (bonus numbers do not matter): 21 × 435 = 9,135 tickets, about 1 in 1,127
• 5th prize, 4 main numbers: 35 × 4,060 = 142,100 tickets, about 1 in 72
• 6th prize, 3 main numbers plus at least one bonus number (3+B): 35 × 6,930 = 242,550 tickets, about 1 in 42

In the 4th and 5th prize rows the remaining numbers on the ticket can be any of the 30 numbers that are not main numbers, so the count uses C(30, 2) = 435 and C(30, 3) = 4,060. For the 6th prize, the other 4 numbers include one bonus number and three undrawn numbers (2 × 3,276 ways) or both bonus numbers and two undrawn numbers (378 ways), which gives 6,930.

Adding the six rows gives 393,996 winning tickets, so about 1 ticket in 26 wins something. Matching 3 main numbers without a bonus number wins nothing; that happens on about 1 ticket in 14.

A bonus number drawn from the same pool cannot be multiplied in the way a separate bonus ball can (see the next section): it simply changes which tickets fall into which tier, while the total number of possible tickets stays at 10,295,472.

Games with a separate bonus ball

Some large lotteries draw main numbers from one pool and an extra ball from a second pool. Because the two choices are independent, you multiply the counts:

• Powerball: 5 white balls from 69, then 1 red ball from 26. C(69, 5) = 11,238,513, and 11,238,513 × 26 = 292,201,338.
• Mega Millions: 5 from 70, then 1 from 24. C(70, 5) = 12,103,014, and 12,103,014 × 24 = 290,472,336.

Game rules change from time to time, so always check the official rules for the current pool sizes. The method stays the same: count each independent choice and multiply.

By hand, spreadsheet or simulation?

By hand

Working through the formula once is the best way to understand where the numbers come from, and it gives the exact answer. It is slow and easy to slip on the multiplication, so check your result.

Spreadsheet or calculator

COMBIN gives exact answers instantly, and you can lay out a whole prize table in a few rows. It does not, however, show what those odds feel like.

Simulation

A simulator draws random tickets over and over and counts what happens. It never gives an exact answer, only an estimate that gets closer with more draws, and long runs take time. Its strength is making a number like 1 in 10 million tangible: you watch hundreds of thousands of draws go by without a jackpot, and you see that two runs can differ wildly. Use the formula for the answer and a simulation for the intuition.

Watching the odds in Lottery Simulator

Lottery Simulator uses the Loto 7 format described above: 7 numbers from 1 to 37, with 7 main and 2 bonus numbers in every draw. Nothing is bought: every "purchase" is a counter in the simulation.

1. Look at Your Selected Numbers, a random ticket of 7 different numbers. Tap New Numbers for a different random ticket; every ticket has the same chance.
2. Tap Buy One to run a single draw. The winning numbers and the 2 bonus numbers appear, your matches are highlighted, and any prize tier is named.
3. Tap Auto Buy to keep drawing automatically, then Stop when you have seen enough.
4. Tap Hit First Prize to simulate draw after draw at high speed until all 7 match. Note how many attempts it took, then Reset All and try again: the counts vary enormously from run to run.
5. Open the Win History table. It lists the six prize tiers with their conditions (7, 6+B, 6, 5, 4 and 3+B), how many times each was hit and its probability, the same figures as the Loto 7 list above. Over a long run, the 4th, 5th and 6th prize counts settle close to those values.

The Simulation Results panel also converts the number of attempts into years of play at one ticket a week, which is often the most striking way to see what 1 in 10,295,472 means.

If gambling stops being fun

Lotteries are designed so that, on average, players get back less than they spend. If you find yourself spending more than you planned, chasing losses or hiding how much you play, free and confidential help is available. In the US, the National Council on Problem Gambling runs a national helpline; in the UK, GamCare runs the National Gambling Helpline. Elsewhere, search for your local gambling helpline.

Frequently asked questions