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?

0.00054

Step-by-step 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

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