Probability basics for lottery players

Probability in one idea

If every outcome is equally likely, probability is simply the number of outcomes you want divided by the total number of outcomes. A fair die has six faces, so the chance of rolling a 4 is 1 in 6, about 16.7%. A lottery works the same way: if there are 292 million possible tickets and yours is one of them, your chance of matching the winning ticket is 1 in 292 million.

Counting combinations

Lottery numbers are drawn without order mattering, so we count combinations, not orderings. The number of ways to choose k items from n is written "n choose k" and calculated as n! / (k! (n−k)!). Choosing 6 from 49 gives 13,983,816 combinations. Choosing 5 from 69 gives 11,238,513. These numbers are why the odds are what they are. Try the lottery odds calculator to see the maths for any game.

Independence

Two events are independent if one does not affect the other. Coin flips are independent: after five heads in a row, the next flip is still 50/50. Lottery draws are independent too: last week's numbers have no influence on this week's. This one idea defeats "hot numbers", "due numbers" and many betting systems. See the gambler's fallacy.

The complement rule and "at least one"

It is usually easier to calculate the chance something does not happen and subtract from 1. If one ticket has a 1.4% chance of winning, it has a 98.6% chance of losing. If you play 10 independent draws with that chance each, your chance of losing every one is 0.986 to the power 10, about 86.8%, so the chance of winning at least once is about 13.2%. This is how "chance over many draws" is worked out.

Expected value

Expected value (EV) multiplies each possible result by its probability and adds them up. If a game pays $10 with probability 0.3 and nothing otherwise, its EV is $3. A ticket that costs $5 and returns an EV of $4.50 loses 50 cents on average. Every lottery has a negative EV, because the operator takes a share. Use the expected value calculator to test a jackpot, and read lottery expected value explained.

Variance and the law of large numbers

Variance describes how spread out results are. Lotteries have enormous variance: most plays lose, a few win a little, and very rarely one wins a lot. The law of large numbers says that over very many plays, your average result will approach the expected value. With only a few plays, anything can happen. That is why a lucky win says nothing about whether a game is worth playing, and an unlucky streak says nothing about whether it is rigged.

A few quick checks you can do

• Ten different Powerball tickets give you ten times the chance of one: 10 in 292 million.
• A 1 in 1,000 event happening once in 1,000 tries is not guaranteed: the chance of at least one is about 63%.
• If someone says a number is "due", ask what mechanism could possibly make it so.
• If a system claims to improve your odds, ask what it changes about the combinations. The answer is nothing.

Example: building the Powerball odds from scratch

1. 1Count the ways to choose 5 white balls from 69: 69×68×67×66×65 divided by 5×4×3×2×1 = 11,238,513.
2. 2Count the ways to choose the Powerball: 26.
3. 3Multiply: 11,238,513 × 26 = 292,201,338 possible tickets.
4. 4Exactly one wins the jackpot, so the chance is 1 in 292,201,338, or about 0.00000034%.
5. 5For the $4 Powerball-only prize: match none of the white balls (C(64,5)/C(69,5) = 7,624,512 / 11,238,513 = 0.678) and the Powerball (1/26 = 0.0385). Multiply to get 0.0261, or about 1 in 38.3.

Where people go wrong

• Confusing "unlikely" with "impossible". Rare events happen when enough people play.
• Thinking a streak changes the next outcome.
• Adding probabilities that should be multiplied, or the reverse.
• Judging a game by its best-case prize instead of its average.