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?
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To calculate the probability of winning a prize, we need to determine the number of favorable outcomes and the total number of possible outcomes.
First, let's find the number of favorable outcomes. There are two ways to win a prize:
1. Match at least two of the white balls.
2. Match the SuperBall.
1. Matching at least two of the white balls:
The number of ways to choose 2 white balls out of 3 is given by the combination formula C(3, 2) = 3. Additionally, there are 18 remaining balls that could be the third ball that doesn't match the white balls. Therefore, the number of favorable outcomes for this case is 3 * 18 = 54.
2. Matching the SuperBall:
The SuperBall can be any one of the ten balls numbered 21-30, so the number of favorable outcomes for this case is 10.
Now, let's find the total number of possible outcomes.
For the first part (matching at least two white balls), we already found 54 favorable outcomes, but there are still more possibilities if all three white balls match. The number of ways to choose 3 white balls out of 3 is given by C(3, 3) = 1. So there is 1 extra favorable outcome for this case. Therefore, the total number of favorable outcomes is 54 + 1 = 55.
For the second part (matching the SuperBall), there are 10 favorable outcomes.
The total number of possible outcomes is the product of the number of choices for each part.
For the white balls, there are C(20, 3) = 1140 possible outcomes.
For the SuperBall, there are 10 possible outcomes.
So the total number of possible outcomes is 1140 * 10 = 11,400.
Finally, to calculate the probability of winning a prize, divide the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 55 + 10 / 11,400
Probability = 65 / 11,400
Probability ≈ 0.0057
Therefore, the probability of winning a prize is approximately 0.0057, or 0.57%.