I have written some articles on the probabilities of winning various lotteries -- Powerball and MegaMillions in the US, 6/49 and LottoMax in Canada, EuroMillions in Western Europe. The answer depends on the structure of the lottery, that is, how many balls are chosen and the range of numbers.
For the Powerball, you pick 5 different numbers from between 1 and 59, and then a "power" ball from between 1 and 35. That gives you a total of
35 x [59 x 58 x 57 x 56 x 55]/[5 x 4 x 3 x 2 x 1] = 175,223,510
different ticket combinations. In the expression above, you divide by 5x4x3x2x1 because the order of the first 5 numbers does not matter. To get the probability of winning the jackpot, you compute
1/175,223,510 = 0.000000005707
which is the same as 0.0000005707%. A very low chance! The size of the jackpot does not affect the probability of winning, only the combinatorial struction of the lottery. If the jackpot was advertised as only a few million, your chances of winning would be the same.
I hope that answers your question. You can see more in my hub
Also, the websites of the different lotteries give you the probabilities of all their prizes, so you don't have to do the math by hand.