Lottery mathematics
Lottery mathematics is used to calculate probabilities of winning or losing a lottery game. It is based primarily on combinatorics, particularly the twelvefold way and combinations without replacement. It can also be used to analyze coincidences that happen in lottery drawings, such as repeated numbers appearing across different draws.
In the following
- is the number of balls in a pool of balls that the winning balls are drawn from, without replacement.
- is the number of winning balls drawn from the pool.
- is the number of balls listed on the lottery ticket.
- is the number of balls that match on the lottery ticket and within the winning set.
Single pool of balls
Suppose there are unique balls from which balls are to be drawn without replacement. Suppose a subset of balls is drawn as the winning set. Suppose a subset of balls is selected on a lottery ticket. Suppose of the balls from the lottery ticket are also among the balls in the winning set. Out of the possible ways to draw the winning set, there are ways to have of them come from the on the lottery ticket and ways to have of them come from the set of not mentioned on the lottery ticket. That is, the probability of getting matches is given by the following formula when there are balls in the pool, each lottery ticket selects balls, and is the number of winning balls drawn for the lottery.Single pool examples
The chances of getting matches when drawing balls from a pool of balls and lottery tickets with balls each:| matches | balls from | balls from |
| 0 | ||
| 1 | ||
| 2 | ||
| 3 | ||
| 4 | ||
| 5 | ||
| 6 |
Power balls from a separate pool of balls
Some lotteries also have one or more power balls drawn from a separate pool of balls. For example, the first drawing may be for balls out of balls and then power ball may be drawn from balls. For both pools, the drawing is without replacement. Similarly a lottery ticket will indicate regular balls and power balls. Often the number of balls on the lottery ticket is the same as the winning set: and. The probabilities of getting matches in the first drawing and matches in the power ball drawing is just the product of the individual probabilities:and similarly for three or more pools of balls.
For example, this formula could be used if one ball is drawn from a pool of balls numbered 0 to 9 and then a second and third ball are also drawn from their own pools of 10 balls:, and. However, this case is more easily modeled as if a single ball, numbered from 0 to 999 is drawn from a single pool of 1000 balls.
Separate pool examples
The chances of getting matches when drawing balls from a pool of balls, getting matches when drawing power balls from a separate pool of balls, and having regular balls and power balls selected on the lottery ticket:| matches | balls from and balls from |
| 0 + 0 | |
| 0 + 1 | |
| 1 + 0 | |
| 1 + 1 | |
| 2 + 0 | |
| 2 + 1 | |
| 3 + 0 | |
| 3 + 1 | |
| 4 + 0 | |
| 4 + 1 | |
| 5 + 0 | |
| 5 + 1 |
Bonus balls from the same pool of balls
Some lotteries have one or more bonus balls drawn from the original pool of balls after the first round of balls is drawn. In this scenario each lottery ticket indicates balls out of possibilities, but the drawing is for balls plus bonus balls, without replacement. The probability that balls from the first drawing match the lottery ticket and balls from the bonus-ball drawing match the lottery ticket is given byExample of balls and bonus balls from the same pool
When of numbers are on a lottery ticket but the winning set is numbers plus bonus ball then probabilities are as follows.| matches | balls on ticket, balls and bonus ball drawn from balls |
| 0 + 0 | |
| 0 + 1 | |
| 1 + 0 | |
| 1 + 1 | |
| 2 + 0 | |
| 2 + 1 | |
| 3 + 0 | |
| 3 + 1 | |
| 4 + 0 | |
| 4 + 1 | |
| 5 + 0 | |
| 5 + 1 | |
| 6 + 0 | |
| 6 + 1 | Impossible |