It is based primarily on combinatorics, particularly the twelvefold way and combinations without replacement.
[1] In a typical 6/49 game, each player chooses six distinct numbers from a range of 1–49.
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
"Combination" means the group of numbers selected, irrespective of the order in which they are drawn.
An eventual 7th drawn number, the reserve or bonus, is presented at the end.
An alternative method of calculating the odds is to note that the probability of the first ball corresponding to one of the six chosen is 6/49; the probability of the second ball corresponding to one of the remaining five chosen is 5/48; and so on.
This yields a final formula of A 7th ball often is drawn as reserve ball, in the past only a second chance to get 5+1 numbers correct with 6 numbers played.
For a score of n (for example, if 3 choices match three of the 6 balls drawn, then n = 3),
For example, one has to buy 13,983,816 different tickets to ensure to win the jackpot in a 6/49 game.
Lottery organizations have laws, rules and safeguards in place to prevent gamblers from executing such an operation.
Further, just winning the jackpot by buying every possible combination does not guarantee that one will break even or make a profit.
the cost for obtaining a ticket (e.g. including the logistics);
The above theoretical "chance to break-even" point is slightly offset by the sum
The payout depends on the number of winning tickets for all the prizes
In addition, in one operation the logistics failed and not all combinations could be obtained.
Another example of such a game is Mega Millions, albeit with different jackpot odds.
Where more than 1 powerball is drawn from a separate pool of balls to the main lottery (for example, in the EuroMillions game), the odds of the different possible powerball matching scores are calculated using the method shown in the "other scores" section above (in other words, the powerballs are like a mini-lottery in their own right), and then multiplied by the odds of achieving the required main-lottery score.
If the powerball is drawn from the same pool of numbers as the main lottery, then, for a given target score, the number of winning combinations includes the powerball.
Since the number of remaining balls is 43, and the ticket has 1 unmatched number remaining, 1/43 of these 258 combinations will match the next ball drawn (the powerball), leaving 258/43 = 6 ways of achieving it.
for the score of 2 multiplied by the probability of one of the remaining four numbers matching the bonus ball, which is 4/43.
, the probability of obtaining the score of 2 and the bonus ball is
lottery with one bonus ball from a separate pool of
lottery with no bonus ball from a separate pool of
It is a hard (and often open) problem to calculate the minimum number of tickets one needs to purchase to guarantee that at least one of these tickets matches at least 2 numbers.
[3] Coincidences in lottery drawings often capture our imagination and can make news headlines as they seemingly highlight patterns in what should be entirely random outcomes.
Lottery mathematics can be used to analyze these extraordinary events.
This makes it easy to calculate quantities of interest from information theory.
For example, the information content of any event is easy to calculate, by the formula
For example, winning in the example § Choosing 6 from 49 above is a Bernoulli-distributed random variable
Oftentimes the random variable of interest in the lottery is a Bernoulli trial.