Lottery game mathematics is used to calculate probabilities of winning or losing a lotto game. It is based greatly on combinatorics, particularly the twelvefold way and mixes without replacement. In a typical 6/49 game, each player chooses 6 distinct numbers from a variety of 1-49. If the 6 numbers on a ticket match the numbers drawn by the lotto, the ticket holder is a prize winner despite the order of the numbers.
The chance of winning can be shown as follows: The first number drawn has a 1 in 49 chance of matching. When the draw concerns the 2nd number, there are now just 48 balls left in the bag, since the balls are drawn without replacement. So there is now a 1 in 48 possibilities of predicting this number.
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This suggests that the possibility of properly predicting 2 numbers drawn from 49 in the right order is computed as 1 in 49 48. On drawing the 3rd number there are just 47 ways of selecting the number; however, obviously we might have gotten to this point in any of 49 48 methods, so the possibilities of correctly predicting 3 numbers drawn from 49, once again in the proper order, is 1 in 49 48 47.
This exercise to 10,068,347,520, which is much bigger than the 14 million mentioned 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.
Dividing 10,068,347,520 by 720 offers 13,983,816, also composed as 49! 6! (49 6)! \ displaystyle 49! \ over 6! *( 49-6)!, or more usually as (n k) = n! k! (n k)! \ displaystyle n \ select k = n! \ over k!( n-k)!, where n is the variety of alternatives and k is the variety of choices.
This function is called the combination function; in Microsoft Excel, this function is carried out as COMBIN( n, k). For example, COMBIN( 49, 6) (the estimation revealed above), would return 13,983,816. For the rest of this short article, we will use the notation (n k) \ displaystyle n \ select k. “Mix” indicates the group of numbers picked, irrespective of the order in which they are drawn.