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?
To win the jackpot, you must match either:
- Two of the three white balls
- The red SuperBall
Let's calculate the probability for each of these cases and then add them up.
Case 1: Matching two of the three white balls
There are \(C(12,3) = 220\) ways to choose three white balls out of 12. And there are \(C(3,2) = 3\) ways to choose two of these three balls. So, the number of favorable outcomes for this case is \(3 * C(12,3) = 660\).
Since there are C(12,3) ways to choose three white balls from your ticket, the probability of matching two of the three white balls is\( 660 / C(12,3) = 660 / 220 = 3 / 10\).
Case 2: Matching the red SuperBall
There are 20 - 13 + 1 = 8 ways to choose the red SuperBall. And there is only 1 way to choose the red SuperBall from your ticket. So, the number of favorable outcomes for this case is 8.
Since there are 20 - 13 + 1 = 8 ways to choose the red SuperBall from your ticket, the probability of matching the red SuperBall is 8 / 8 = 1.
Adding the probabilities for these two cases, we get the total probability of winning the jackpot:
P = 3/10 + 1 = 7/10
So, the probability of winning the jackpot is 7/10.