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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?

 Dec 13, 2020
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We compute the complement: we'll count the number of losing tickets.

We saw in part (a) that there are 1200 total possibilities.

To have a losing ticket, you must have at most one correct white ball, and miss the SuperBall.

You miss all 3 white balls if your ticket contains 3 of the 7 white numbers that were not drawn, so there are  possibilities.

You hit 1 white ball and miss the others if your ticket contains 1 of the 3 white numbers that were drawn and 2 of the 7 white numbers that were not drawn, so there are  possibilities.

You miss the SuperBall if you have one of the 9 red numbers that were not drawn.

Therefore, there are  losing tickets.

Hence, there are  winning tickets, and your probability of winning a super prize is

Note: We can approach this problem using direct counting, but there are a number of cases: 1) Matching 3 white balls with any super ball. 2) Matching 2 white balls with any super ball. 3) Matching 1 white ball and matching the super ball. 4) Matching 0 white balls and matching the super ball.

Given the numerous cases here, a complementary counting approach is a faster approach.

 Jan 25, 2021

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