Picking 6 numbers out of 30, where you make exactly 6 picks to get those 6 winning numbers, where all 30 possible numbers are different, picked numbers can't be repeated, and the order in which the numbers are picked doesn't matter:
Buying 1 ticket:
P(first picked number is a winner) = 6/30
P(2nd picked number is a winner) = 5/29
P(3rd picked number is a winner) = 4/28
P(4th picked number is a winner) = 3/27
P(5th picked number is a winner) = 2/26
P(6th picked number is a winner) = 1/25
P(all 6 of the above events happen) = 6/30 * 5/29 * 4/28 * 3/27 * 2/26 * 1/25
P(all 6 of the above events happen) = (6 * 5 * 4 * 3 * 2 * 1) / (30 * 29 * 28 * 27 * 26 * 25)
P(all 6 of the above events happen) = 24! * 6! / 30!
P(all 6 of the above events happen) = "30 choose 6"
P(all 6 of the above events happen) = 30 nCr 6 (on a calculator)
P(all 6 of the above events happen) = 0.00000168413 (about 1 in 600,000)
Buying 100 tickets and you want to win at least once:
P(at least 1 win) = 1 - (1 - 0.00000168413)^100
P(at least 1 win) = 0.00016839896 (about 1 in 5,938.3)
Buying 100 tickets and you want to win exactly once (not twice or more):
P(win exactly once) = P(win on a single draw) * ( P(lose on a single draw) )^99 * (# ways to arrange 1 win and 100 losses)
P(win exactly once) = 0.00000168413 * (1 - 0.00000168413)^99 * 100
P(win exactly once) = 0.00016838492 (about 1 in 5,938.8)