Question

In a​ lottery, the top cash prize was ​$693 ​million, going to three lucky winners. Players pick four different numbers from 1 to 51 and one number from 1 to 48. A player wins a minimum award of $ 400 by correctly matching two numbers drawn from the white balls​ (1 through 51​) and matching the number on the gold ball​ (1 through 48​). What is the probability of winning the minimum​ award?

Answer 1

0.00054

Explanation:

Gold ball: we have one chance of choosing the correct ball from 1 to 48, so the probability is 1/48

White balls: we will pick 4 numbers from 1 to 51, so the total possibilities of groups of numbers we can choose is a combination of 51 choose 4:

C(51, 4) = 51! / (4! * 47!) = 51*50*49*48/(4*3*2)

If we want to correctly guess two numbers, we have groups of 2 winning numbers in the 4 numbers we guess, so a combination of 4 choose 2, and we also have to incorrectly guess the other 2 winning numbers in the 47 remaining numbers we didn't choose, so a combination of 47 choose 2:

C(4, 2) = 4! / (2! * 2!) = 4*3/2 = 6

C(47, 2) = 47! / (2! * 45!) = 47*46/2

The probability of the white balls case is the two combinations above over the total number of cases (the first combination we calculated):

P = C(4, 2) * C(47, 2) / C(51, 4)

P = (6 * 47 * 46/2) / (51 * 50 * 49 * 48/(4*3*2))

P = (3 * 47 * 46 * 4 * 3 * 2) / (51 * 50 * 49 * 48) = 0.026

Then the final probability is the probability in the gold ball case times the probability in the white balls case:

P = (1/48) * (3 * 47 * 46 * 4 * 3 * 2) / (51 * 50 * 49 * 48) = 0.00054