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 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?
The probability of winning a super prize is the sum of the probability of matching at least two white balls and the probability of matching the red SuperBall.
The probability of matching the red SuperBall is 1/8, as there are 8 red balls and only 1 is drawn.
The probability of matching at least two white balls is a bit more complex to calculate. To match at least two white balls, we can either match exactly two or exactly three.
The probability of matching exactly two white balls is: (3 choose 2) * (12-3)/(12-3) * (12-2)/(12-2) = 3 * 11/12 * 10/11 = 30/44
The probability of matching exactly three white balls is: (3 choose 3) * (12-3)/(12-3) * (12-2)/(12-2) * (12-1)/(12-1) = 1 * 11/12 * 10/11 * 9/10 = 9/22
So, the probability of matching at least two white balls is: P(matching at least 2 white balls) = P(matching exactly 2 white balls) + P(matching exactly 3 white balls) = 30/44 + 9/22 = 39/44
Therefore, the probability of winning a super prize is: P(winning a super prize) = P(matching at least 2 white balls) + P(matching the red SuperBall) = 39/44 + 1/8 = 39/44 + 2/44 = 41/44
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.
