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?
+53 There are $\binom{12}{3} = 220$ ways to choose three numbers from the white balls, and $\binom{20}{1} = 20$ ways to choose one number from the red balls. Hence, the total number of possible outcomes when you buy a ticket is $220 \cdot 20 = 4400.$
For you to win the super prize, you need either:
Two of your white balls to match the three white balls drawn, which can be achieved in $\binom{3}{2} = 3$ ways, or
One of your white balls to match one of the three white balls drawn, and your red ball to match the red ball drawn, which can be achieved in $3 \cdot 1 = 3$ ways.
All three of your white balls to match the three white balls drawn, which can be achieved in $\binom{3}{3} = 1$ way.
So the number of favorable outcomes is $3 \cdot \binom{3}{2} + 3 \cdot 1 + 1 = 10.$
The probability of winning a super prize is 10/4400 = 1/440.
