Lottery Odds Calculator

About the Lottery Odds Calculator

Lottery jackpots are advertised in huge, eye-catching numbers; the odds of winning them are printed in small type, if at all. Those odds are enormous because of how quickly the number of possible combinations grows. Choosing 6 numbers from 49 sounds manageable, but there are almost 14 million ways to do it, and only one of them wins the jackpot.

This lottery odds calculator works out the odds of winning the jackpot for any lottery format: the numbers picked from a pool, plus any bonus numbers drawn from a separate pool, as in Powerball or EuroMillions. It shows the number of combinations, your chance with several tickets, the odds of matching fewer numbers, and puts the odds in perspective.

How to Use the Lottery Odds Calculator

Enter the numbers picked and the size of the pool they come from, such as 6 from 49 or 5 from 69.

If the lottery draws bonus numbers from a separate pool, enter how many and the bonus pool size: 1 from 26 for Powerball, 2 from 12 for EuroMillions. Leave both at 0 otherwise.

Enter the number of different tickets you buy.

How Lottery Odds Are Worked Out

  combinations  =  C(pool, picked) × C(bonus pool, bonus picked)
  C(n, k)       =  n! ÷ (k! × (n − k)!)
  jackpot odds  =  1 in combinations
  j matches     =  C(picked, j) × C(pool − picked, picked − j) ÷ C(pool, picked)

Step-by-Step Example

A 6/49 lottery.

  Combinations:  49 × 48 × 47 × 46 × 45 × 44 ÷ 720   =  13,983,816
  Jackpot:       1 in 13,983,816
  5 numbers:     C(6,5) × C(43,1) ÷ 13,983,816        =  1 in 54,201
  4 numbers:     C(6,4) × C(43,2) ÷ 13,983,816        =  1 in 1,032
  3 numbers:     C(6,3) × C(43,3) ÷ 13,983,816        =  1 in 57

Odds of Well-Known Lotteries

Bonus balls multiply the odds: EuroMillions' 2 Lucky Stars from 12 add 66 combinations to the 2,118,760 ways of picking 5 main numbers from 50.

Putting the Odds in Perspective

Numbers in the millions are hard to picture. Buying one 6/49 ticket a week, you would expect to wait around 269,000 years for a jackpot. You are far more likely to be struck by lightning in your lifetime than to win a major jackpot with a single ticket. That does not make playing wrong — for many people it is inexpensive entertainment and a moment of hope — but it is wise to spend only what you would happily spend on any other entertainment.

Does Buying More Tickets Help?

Each different ticket covers one more combination, so ten different tickets are ten times as likely to win as one. But ten times a tiny chance is still tiny: 10 tickets for a 6/49 draw give odds of about 1 in 1.4 million. Syndicates pool money to buy many tickets and share any prize, which raises the chance of winning something while dividing the reward.

Choosing Numbers

Every combination is exactly as likely as any other, including 1, 2, 3, 4, 5, 6. Past draws do not affect future ones, so "hot" and "cold" numbers mean nothing. What your choice can affect is how many people you might share a jackpot with: many players choose birthdays, so numbers from 1 to 31 are more popular. Picking random or higher numbers does not improve your odds, but can increase your share if you win.

Expected Value of a Ticket

The expected value of a lottery ticket is the average amount it returns: each prize multiplied by its chance of being won, added together. For most lotteries, roughly half of ticket sales go back as prizes, so the expected value of a ticket is about half its price, even allowing for the jackpot. Very large rollover jackpots can raise the expected value, but the chance of sharing the jackpot with other winners also rises when many more tickets are sold, and taxes may reduce the prize. Lotteries are designed to raise money, so over time players as a group always get back less than they pay in.

Smaller Prizes

Most lotteries pay smaller prizes for matching fewer numbers, and these are where most winning tickets come from. In a 6/49 game, matching 3 numbers is about 1 in 57, and many games add extra prizes for matching the bonus ball with fewer main numbers. The calculator's fewer-matches line covers the main numbers; check your lottery's prize table for the rules on bonus balls.

Scratch Cards and Other Games

Scratch cards and instant-win games work differently from draw lotteries: a fixed number of winning cards is printed in each batch, and the overall odds of winning any prize are printed on the back, often around 1 in 3 or 1 in 4, though most wins only return the price of the card.

Understanding Your Result

Jackpot odds gives the chance of winning the top prize.

Combinations shows the number of possible tickets.

Your chance gives the chance with the tickets you buy.

Fewer matches gives the odds of matching fewer main numbers.

In perspective shows how long a weekly player would wait on average.

Worth knowing explains why every combination is equally likely.

When Should You Use This Calculator?

Comparing lottery games.

Understanding the real odds before you play.

Teaching combinations and probability.

Deciding on a syndicate or number of tickets.

Checking advertised odds.

Common Mistakes

Adding the bonus pool instead of multiplying.

Thinking some numbers are due.

Buying duplicate tickets, which add no chance.

Assuming more tickets make a jackpot likely.

Spending more than you can afford.