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# Probability Question

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In the SuperLottery, three balls are drawn (at random, without replacement) from white balls numbered from \(1\) to \(12\) and one SuperBall is drawn (at random) from red balls numbered from \(13\) to \(20\) When you buy a ticket, you choose three numbers from \(1\) to \(12\) and one number from \(13\) 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?

Please help ASAP... I know this is done with casework, but I just did like 20 other questions like this, so my brain is fried...

Dec 3, 2023

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There are (312​)=220 ways to choose the three white balls on your ticket. For each of these choices, there is one way to choose the red SuperBall. There is also a way to choose the three white balls and the red SuperBall in the same order that they were drawn, so there are also (312​) winning ticket combinations in which your ticket matches at least two of the white balls drawn.

Finally, there are 8 winning ticket combinations in which your ticket matches the red SuperBall, but does not match any of the white balls drawn. Therefore, there are a total of 220+220+8=448 winning ticket combinations.

Since there are a total of (312​)⋅8=840 ways to choose three white balls and one red ball, the probability that your ticket is a winning ticket is 448/840 = 56/105​​.

Dec 3, 2023