+355 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?
+1471 To find the probability of winning a super prize in the SuperLottery, we need to consider two mutually exclusive events:
1. **Winning by matching at least two of the white balls.**
2. **Winning by matching the red SuperBall.**
We'll calculate the probabilities for each event and then add them together.
1. **Winning by matching at least two of the white balls:**
To calculate this probability, we can find the probability of not matching any of the white balls and subtract it from 1.
The probability of not matching any of the white balls on a single draw is:
\[\frac{{9 \choose 3}}{{12 \choose 3}}\]
So, the probability of matching at least two of the white balls is:
\[1 - \frac{{9 \choose 3}}{{12 \choose 3}}\]
2. **Winning by matching the red SuperBall:**
The probability of matching the red SuperBall is simply \( \frac{1}{8} \) since there's only one SuperBall drawn from 8 possibilities.
Now, let's calculate these probabilities:
1. Probability of winning by matching at least two of the white balls:
\[1 - \frac{{9 \choose 3}}{{12 \choose 3}} = 1 - \frac{84}{220} = 1 - \frac{21}{55} = \frac{34}{55}\]
2. Probability of winning by matching the red SuperBall:
\[P(\text{Red SuperBall}) = \frac{1}{8}\]
Finally, to find the total probability of winning a super prize, we add the probabilities of the two mutually exclusive events:
\[P(\text{Winning super prize}) = P(\text{White balls}) + P(\text{Red SuperBall}) = \frac{34}{55} + \frac{1}{8}\]
\[= \frac{272}{440} + \frac{55}{440}\]
\[= \frac{272 + 55}{440}\]
\[= \frac{327}{440}\]
So, the probability of winning a super prize in the SuperLottery is \( \frac{327}{440} \).
