Lottery mathematics
Lottery mathematics is used to calculate probabilities in a lottery game.
In a typical 6/49 game, each player chooses six non-duplicate numbers from a range of 1-49. If the six numbers on a ticket match the numbers drawn by the lottery, the ticket holder is a jackpot winner—regardless of the order of the numbers. The probability of this happening is 1 in 13,983,816.
The chance of winning can be demonstrated as follows:
The first number drawn has a 1 in 49 chance of matching . When the draw comes to the second number, there are now only 48 balls left in the bag (because the balls already drawn are not returned to the bag) so there is now a 1 in 48 chance of predicting this number.
Thus for each of the 49 ways of choosing the first number there are 48 different ways of choosing the second. This means that the probability of correctly predicting 2 numbers drawn from 49 in the correct order is calculated as 1 in 49 × 48. On drawing the third number there are only 47 ways of choosing the number; but of course we could have arrived at this point in any of 49 × 48 ways, so the chances of correctly predicting 3 numbers drawn from 49, again in the correct order, is 1 in 49 × 48 × 47. This continues until the sixth number has been drawn, giving the final calculation, 49 × 48 × 47 × 46 × 45 × 44, which can also be written as or 49 factorial divided by 43 factorial. This works out to 10,068,347,520, which is much bigger than the ~14 million stated above.
However, the order of the 6 numbers is not significant. That is, if a ticket has the numbers 1, 2, 3, 4, 5, and 6, it wins as long as all the numbers 1 through 6 are drawn, no matter what order they come out in. Accordingly, given any set of 6 numbers, there are 6 × 5 × 4 × 3 × 2 × 1 = 6! or 720 orders in which they could be drawn. Dividing 10,068,347,520 by 720 gives 13,983,816, also written as 49! / (6! × (49 - 6)!), or more generally as
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