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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?
PS IMPORTANT: PLEASE EXPLAIN HOW YOU GOT TO THE ANSWER/PROCESS!
EZ.
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 \(\binom{7}{3} = \dfrac{7 \cdot 6 \cdot 5}{6} = 35 \)
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 \(3\binom{7}{2} = \dfrac{3 \cdot 7 \cdot 6}{2} = 63 \).
You miss the SuperBall if you have one of the 9 red numbers that were not drawn.
Therefore, there are \((35 + 63) \cdot 9 = 882 \) Hence, there are \((35 + 63) \cdot 9 = 882 \) winning tickets, and your probability of winning a super prize is \(\frac{318}{1200} = \boxed{\frac{53}{200}}. \)
