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Stuck on a problem...

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PROBLEM:

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?

MY THOUGHTS:

I tried thinking about the ways that you could NOT win a super prize.  This happens when you get the wrong red SuperBall AND get either 1 or 0 correct white balls.  There is a 9/10 chance of getting the wrong red SuperBall, and I am currently stuck on the chance of getting 1 or 0 correct white balls.  Please help.  I will try to work with you, and I am not trying to directly get the answer, just some hints.  Thanks in advance!

Feb 7, 2021

#1
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Something else I thought of:

For the white balls, you just need to get 2 wrong to not get the super prize.  To get 1 wrong is 9/10.  To get 2 wrong is 81/100.  Is there a 9/10*81/100 chance of not getting a super prize or did I make a mistake somewhere?

Feb 7, 2021
#2
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Case 1: Superball

The probability is 1/10, because you choose 1 out of 10.

Case 2: 2 whites

The probability is 1/C(10,2) = 1/45, because you choose 2 out of 10.

Case 3: 3 whites

The probability is 1/C(10,3) = 1/120, because you choose 3 out of 10.

Adding these, you get 1/10 + 1/45 + 1/120 = 47/360.

Feb 13, 2021