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
P is the number of balls in a pool of balls that the winning balls are drawn from, without replacement. W is the number of winning balls drawn from the pool. T is the number of balls listed on the lottery ticket. (It often equals W.) M is the number of balls that match; these balls are both on the lottery ticket and within the winning set.
Single pool of balls Suppose there are P unique balls (such as P = 49) from which balls are to be drawn without replacement. Suppose a subset of W balls (such as W = 6) is drawn as the winning set. Suppose a subset of T balls (such as T = 6) is selected on a lottery ticket. Suppose M of the T balls from the lottery ticket are also among the W balls in the winning set. Out of the ( P W ) {\displaystyle {P \choose W}} possible ways (see binomial coefficient) to draw the winning set, there are ( T M ) {\displaystyle {T \choose M}} ways to have M of them come from the T on the lottery ticket and ( P − T W − M ) {\displaystyle {P-T \choose W-M}} ways to have W − M of them come from the set of P − T not mentioned on the lottery ticket. That is, the probability of getting M matches is given by the following formula when there are P balls in the pool, each lottery ticket selects T balls, and W is the number of winning balls drawn for the lottery.
Pr [ M ∣ P , T , W ] = ( T M ) ( P − T W − M ) ( P W ) . {\displaystyle \Pr[M\mid P,T,W]={\frac {{T \choose M}{P-T \choose W-M}}{P \choose W}}\,.}
This can also be solved from the opposite point of view. There are ( P T ) {\displaystyle {P \choose T}} ways to fill out the ticket, ( W M ) {\displaystyle {W \choose M}} ways to choose the matching numbers from winning set, and ( P − W T − M ) {\displaystyle {P-W \choose T-M}} ways to list the remaining numbers in ticket from among the non-winning numbers.
Pr [ M ∣ P , T , W ] = ( W M ) ( P − W T − M ) ( P T ) . {\displaystyle \Pr[M\mid P,T,W]={\frac {{W \choose M}{P-W \choose T-M}}{P \choose T}}\,.}
Both expressions give the same result.
Single pool examples The chances of getting M matches when drawing W balls from a pool of P balls and lottery tickets with T balls each:
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