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Probability

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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 two of the three white balls or the red SuperBall, then you win the jackpot. (You don't need to match the white balls in order). What is the probability that you win the jackpot?

Mar 4, 2023

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We can use the principle of inclusion-exclusion to calculate the probability of winning the jackpot.

First, let's calculate the probability of matching the red SuperBall. There is only one SuperBall, so the probability of matching it is 1/8.

Next, let's calculate the probability of matching exactly two of the three white balls. There are (12 choose 2) = 66 ways to choose two white balls out of 12. For each choice of two white balls, there is exactly one way to choose the third white ball that does not match the ticket. Therefore, there are 66 ways to match exactly two white balls. The probability of matching exactly two white balls is:

(66 / (12 choose 3)) x (2 / 3) x (1 / 2) = 11 / 110

where we have multiplied by (2/3) to account for the probability that the third white ball does not match the ticket, and by (1/2) to account for the fact that the order of the two matching white balls does not matter.

Finally, let's calculate the probability of matching both the SuperBall and exactly two white balls. By the multiplication principle, this probability is:

(1/8) x (11/110) = 11/880

However, we have counted the case where all three white balls match the ticket twice: once as matching the SuperBall and exactly two white balls, and once as matching all three white balls. Therefore, we need to subtract the probability of matching all three white balls:

(1 / (12 choose 3)) x (1 / 3) = 1 / 220

where we have multiplied by (1/3) to account for the probability that the first white ball matches the ticket, and by (1 / (12 choose 3)) to account for the number of ways to choose three white balls out of 12.

Therefore, the probability of winning the jackpot is:

(11/880) - (1/220) = 5/352

Therefore, the probability of winning the jackpot is 5/352.

Mar 5, 2023