It's "relatively" simple. The probabilities follow the Binomial Distribution if you assume the numbers for every ticket are chosen independently and randomly. (That is the case for a "Quick Pick".) The probability of an individual random ticket winning Mega Millions is 1 in 302575350, for example. Powerball is slightly higher probability.
Let's call that probability (1 in 302575350) "p". P (capital letter) is the probability of "x" winners, where x = 0, 1, 2, 3, ..., N, where N is the number of tickets sold.
Then P(x) = C(N,x) * [p^x] * [(1-p)^(1-x)].
"^" means "to the power of".
C(N,x) means the number of combinations of x items chosen from N. C(N,x) = N!/[x!(N-x)!]
If you sum P(x) for every x from 0 to N you will get 1. That is, the number of possible winners is the range of 0 to N.
The complication is that not all tickets are generated this way. People fill in play slips and choose unique combinations for each play, i.e. with no possibility of repeats. So a person who plays M tickets (whatever M is) has M / 302575350 probability of hitting the jackpot. That changes the probabilities of multiple winners. M could all over the place, i.e. it's some frequency distribution. I have never seen this distribution made public by any lottery commissions, so the exact probabilities remain rather elusive, but I think you should grasp the point.
