How Lottery Odds Work

In most lotteries you choose a set of numbers, and the set wins no matter what order the balls come out in. Mathematicians call this a combination. The number of ways to choose k numbers from a pool of n is written C(n, k), and you get it by multiplying n x (n - 1) x ... down k steps, then dividing by k x (k - 1) x ... x 1.

Take the classic 6/49 game. Choosing six from 49 in order would give 49 x 48 x 47 x 46 x 45 x 44 possible sequences. Each set of six can be arranged in 6 x 5 x 4 x 3 x 2 x 1 = 720 orders, and all of those orders are the same ticket, so we divide by 720. The result is 13,983,816 different tickets, which is why the jackpot odds for Lotto 6/49 are 1 in 13,983,816.

Every one of those combinations is exactly as likely as any other. A line such as 1, 2, 3, 4, 5, 6 looks special to us, but the draw machine treats it the same as any jumble of numbers.

Many big games add a second number drawn from its own pool. Powerball has five main numbers from 1 to 69 and one Powerball from 1 to 26. There are 11,238,513 ways to pick five from 69, and the Powerball can be any of 26 values, so the total is 11,238,513 x 26 = 292,201,338 possible tickets.

Mega Millions works the same way with five from 70 and one from 24, giving 290,472,336. EuroMillions draws five from 50 (2,118,760 ways) and two Lucky Stars from 12 (66 ways), so its jackpot odds are 2,118,760 x 66 = 139,838,160. Because the second pool is separate, your bonus number can match one of your main numbers.

Some games draw an extra ball from the same drum as the main numbers. In UK Lotto six main balls come from 59, so the jackpot is 1 in 45,057,474. A seventh bonus ball is then drawn from the 53 that remain. It only matters for the 5 + bonus tier: you need five of the main numbers, and the bonus must be your sixth number. There are six ways to choose which of your numbers is left over, which gives odds of 6 in 45,057,474, or 1 in 7,509,579.

Operators also publish the odds of winning any prize at all. For Powerball that figure is about 1 in 24.9, because lower tiers such as matching just the Powerball are far more likely. Those smaller prizes are often worth less than or close to the ticket price, so the any-prize figure says little about whether a ticket is good value.

A fair lottery machine has no memory. Last week's numbers have no effect on this week's, so a number that has not come up for months is not due, and a number that came up twice is not hot. Believing otherwise is called the gambler's fallacy, and it is one of the most common mistakes people make with chance.

The same logic covers buying the same line every week. Your odds on each draw are unchanged however long you have played. Over many draws your chance of winning at least once grows only in proportion to how many tickets you buy, and so does what you spend.

Expected value is the average amount a ticket returns if you could play the same draw an enormous number of times. Lotteries are designed so that only part of the money taken in is paid back as prizes, with the rest going to costs, good causes or tax. That means the expected value of a ticket is lower than its price in almost every draw, and buying more tickets increases your expected loss rather than your expected gain.

Big numbers are hard to picture. With odds of 1 in 292 million, buying one Powerball line for every draw for the rest of your life would still leave a jackpot extremely unlikely. A useful way to think about a ticket is as entertainment with a known cost, not as a plan or an investment.