The lotto games are covered by the numerical sets named COMBINATIONS. The sets and combinations are very well covered at my website, including unique software: Combinatorics, Permutations
The fundamental formulas of probability are presented here: Probability Theory
The most comprehensive formula to calculate the probability (or odds) of lotto games is known as hypergeometric distribution probability. It covers multiple prizes, not only the jackpot (the first prize in a lotto game).
Here is a text format of the hypergeometric formula for the case ‘m of k in s from n’:
P(m of k in s from n) = ----------------------------
C(s, m) * C(n-s, k-m)
An example: ‘4 of 6 in 12 from 49’. You want to know the odds of winning 4 numbers of the 6 drawn by the lottery in a 49-number game. S=12 is a pool of numbers; e.g. you play a system or lotto wheel which comprises 12 numbers. The default for s is 6 for lotto 6 games. It is the general case when you don’t play lotto wheels or pools of numbers.
C() represents the formula of combinations:
C(n, k) = -------------------------------------
For k=6 and n=49 (the most common lotto game), the formula reads:
C(49, 6) = [49 * 48 * 47 * 46 * 45 * 44] / [1 * 2 * 3 * 4 * 5 * 6]
Euromillions (like the Powerball) is a two-in-one lotto game. You calculate first the odds for the first part (game): the 5 regular numbers. Then, you calculate for the second part: The Star numbers. Finally, you multiply the two results and get the final odds for Euromillions.
I wrote the most comprehensive software for calculating odds and probability and also analyze combinations and lexicographical order. The main software titles: OddsCalc.EXE
A different member (Curlyben) already pointed to my web page that deals specifically with the game of Euromillions. The page also runs two (free) ActiveX controls that calculate the odds and also generate Euromillions combinations (1000 at a time).
Best of luck!