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In the SuperLottery, three balls are drawn (at random) from ten white balls numbered from 1 to 10, and one SuperBall is drawn (at random) from ten red balls numbered from 11 to 20. When you buy a ticket, you choose three numbers from 1 to 10, and one number from 11 to 20. If the numbers on your ticket match at least two of the white balls or match the red SuperBall, then you win a super prize. What is the probability that you win a super prize?

May 23, 2020

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You can win in any of four ways:

1) Getting two of the three winning white balls and the red SuperBall.

Two of the three wining white balls and one out of seven losing white balls:

3C2·7C1 / 10C3  =  3·7/120  =  7/40

Combine this with getting the red SuperBall  =  1/10     --->     7/400

2) Getting two of the three white balls and not getting the red SuperBall:

The  7/40  from above combined with not getting the red SuperBall  =  9/10   --->   63/400

3) Getting all three of the white balls and the red SuperBall:

Three of three winning white balls and none of the seven losing white balls:

3C3·7C0 / 10C3  =  1·1/120  =  1/120

Combine this with getting the red SuperBall  =  1/10     --->     1/200

4)  Getting all three of the white balls and not getting the SuperBall:

The  1/120  from above combined with not getting the red SuperBall = 9/10   --->   3/400