Counting possibilities and measuring chance
How many ways can you pick 5 numbers out of 69? What are the real odds of hitting the Powerball jackpot? How likely is it to roll at least one six in four tries? Questions like these come down to counting combinations and applying a few probability rules. This calculator handles the counting and the arithmetic, including numbers far too large to work out by hand.
The formulas
- Combinations (order does not matter): C(n, k) = n! ÷ (k! × (n − k)!).
- Permutations (order matters): P(n, k) = n! ÷ (n − k)!.
- Combinations with repetition: C(n + k − 1, k).
- Lottery odds: C(n, k) × the number of outcomes of a separate bonus draw.
- At least once in t independent tries: 1 − (1 − p)ᵗ.
- Every time: pᵗ. Expected number of occurrences: p × t.
The exclamation mark is the factorial: 5! = 5 × 4 × 3 × 2 × 1 = 120.
Worked example: Powerball
A Powerball ticket picks 5 white balls from 69 and 1 red ball from 26.
- White balls: C(69, 5) = 69 × 68 × 67 × 66 × 65 ÷ 120 = 11,238,513 combinations.
- With the red ball: 11,238,513 × 26 = 292,201,338.
- The jackpot odds are 1 in 292,201,338. Buying 100 tickets a week for 50 years, about 260,000 tickets, would still give you less than a 0.1 percent chance.
- If order mattered, there would be P(69, 5) = 1,348,621,560 arrangements of the white balls, 120 times more.
The chance of at least one six
With p = 1/6 per roll:
| Rolls | At least one six | Expected sixes |
|---|---|---|
| 1 | 16.67% | 0.17 |
| 2 | 30.56% | 0.33 |
| 4 | 51.77% | 0.67 |
| 6 | 66.51% | 1.00 |
| 10 | 83.85% | 1.67 |
| 20 | 97.39% | 3.33 |
Even after six rolls, when you expect one six on average, there is a one in three chance of seeing none. The same logic explains why a 1 percent risk per year adds up to a 26 percent chance over 30 years.
Combinations or permutations?
Ask whether rearranging the same items creates a different outcome. A poker hand, a lottery ticket or a committee is a combination: the same five cards are the same hand. A race finish, a PIN code or the order of speakers is a permutation. Choosing a president, a vice president and a treasurer among 10 people gives P(10, 3) = 720 possibilities, while choosing a three-person committee gives only C(10, 3) = 120.
Other lotteries and games
- Mega Millions: 5 of 70 and 1 of 24, so C(70, 5) × 24 = 290,472,336.
- Poker: C(52, 5) = 2,598,960 possible five-card hands.
- Ice cream: choosing 3 scoops from 10 flavors, repeats allowed, gives C(12, 3) = 220 combinations.
The gambler’s fallacy
Independent events have no memory. After ten heads in a row, the next coin flip is still 50-50, and a lottery number that has not been drawn for months is no more likely to come up next. The formula for “at least once” works precisely because each try is independent of the others.