What Are the Odds of Winning the Lottery?

Lottery advertising talks about jackpots, rarely about odds. Yet the odds of any draw-based lottery can be worked out with one counting formula, a pen and a little patience. No inside knowledge is required. The numbers below are calculated directly from how the games are built, so you can check every step yourself.

The counting formula

In a typical lottery you choose a set of numbers, and the order in which they are drawn does not matter. The number of possible tickets is therefore the number of combinations of k numbers chosen from n, written C(n, k):

C(n, k) = n × (n − 1) × … × (n − k + 1) ÷ (k × (k − 1) × … × 1)

Each possible combination is equally likely to be drawn. With one ticket, your chance of the jackpot is one in C(n, k).

Example 1: pick 6 numbers from 49

Many national lotteries have used a 6-from-49 matrix. Working it out:

• Top line: 49 × 48 × 47 × 46 × 45 × 44 = 10,068,347,520
• Bottom line: 6 × 5 × 4 × 3 × 2 × 1 = 720
• Result: 10,068,347,520 ÷ 720 = 13,983,816

So a single ticket has a 1 in 13,983,816 chance of matching all six numbers. To picture it: buying one ticket a week, it would take 13,983,816 weeks to cover every combination once. That is a little under 269,000 years.

Example 2: five numbers plus a bonus ball

Some large jackpot games draw two sets of numbers. Powerball, for example, has used a format since 2015 where you pick five numbers from 69 and one extra number from a separate pool of 26. You need both parts right for the top prize, so the two counts are multiplied:

• C(69, 5) = 69 × 68 × 67 × 66 × 65 ÷ 120 = 11,238,513
• Bonus ball: 26 possibilities
• Jackpot odds: 11,238,513 × 26 = 1 in 292,201,338

Game formats change from time to time, so always check the current matrix on the official rules page of the game you are playing and plug those numbers into the same formula.

Smaller prizes

Lower prize tiers are easier to hit, and they can be calculated the same way. In a 6-from-49 game, matching exactly three numbers means choosing 3 of the 6 winning numbers and 3 of the 43 losing ones:

Each figure in the last column is 13,983,816 divided by the number in the middle column. Real games may add bonus-ball tiers, which change these numbers slightly.

What changes your odds, and what does not

More tickets

Ten different lines give you ten chances in 13,983,816 instead of one. That is ten times better and still extremely small. The cost rises in exact step with the improvement.

Lucky, hot or overdue numbers

Draws have no memory. A number that came up last week is no more or less likely to appear this week, and a combination like 1-2-3-4-5-6 is exactly as likely as any other.

Sharing a jackpot

Your choice of numbers cannot improve the odds of winning, but it can affect how many people you share with. Many players choose birthdays, which only go up to 31, so popular numbers are more likely to be split among several winners.

Why it still pays out to someone

Huge jackpots are won because huge numbers of tickets are sold, not because any single ticket has a decent chance. Only part of the money from ticket sales goes back out as prizes, so over time players as a group get back less than they spend. That is the lottery's built-in margin, the same idea behind the progressive jackpots we covered in the allure of progressive jackpots. For converting other kinds of odds into percentages, see making sense of odds.

A sensible way to play

A lottery ticket buys a small daydream, not a financial plan. Tickets are sold only to adults (the minimum age is 18 in most places and higher in a few), and the healthiest approach is a fixed weekly amount you would not miss, with no extra lines bought to chase a big rollover.