In the SmallState Lottery, three white balls are drawn (at random) from twenty balls numbered 1 through 20, and a blue SuperBall is drawn (at random) from ten balls numbered 21 through 30. When you buy a ticket, you select three numbers from 1-20 and one number from 21-30. To win a prize, the numbers on your ticket must match at least two of the white balls or must match the SuperBall.
If you buy a ticket, what is your probability of winning a prize?
To calculate the probability of winning a prize in the SmallState Lottery, we need to consider two separate cases: matching at least two white balls or matching the SuperBall.
Case 1: Matching at least two white balls:
There are 20 white balls in the lottery, and when three are drawn randomly, there are a total of 20C3 (combinations) possible outcomes. This can be calculated as:
20C3 = (20!)/(3!(20-3)!) = (20x19x18)/(3x2x1) = 1140
Out of these 1140 possible outcomes, the winning outcomes are when you have at least two matching white balls. There are three ways this can occur:
1. Match all three white balls: This can occur in 1 way.
2. Match two white balls: This can occur in 3C2 ways, which is 3.
3. Match only one white ball: This can occur in 3C1 ways, which is 3.
So, the total number of winning outcomes in this case is 1 + 3 + 3 = 7.
Therefore, the probability of winning by matching at least two white balls is 7/1140.
Case 2: Matching the SuperBall:
There are 10 blue balls in the lottery, and when one is drawn randomly, there are a total of 10 possible outcomes. The winning outcome in this case occurs only if you match the SuperBall, and this can occur in 1 way.
So, the probability of winning by matching the SuperBall is 1/10.
Finally, to get the probability of winning a prize, we add the probabilities from both cases:
Probability = (7/1140) + (1/10) = 0.0061 + 0.1 = 0.1061 or approximately 10.61%.
Therefore, if you buy a ticket, your probability of winning a prize in the SmallState Lottery is approximately 10.61%.