+311 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?
+585 We can solve this problem by considering the two ways to win and summing their probabilities:
Matching at least two white balls:
Favorable outcomes: We need to calculate the number of winning tickets where we match at least two of the chosen white balls.
Overcounting: We initially consider all possibilities where we pick three white balls (220 ways) and any red ball (8 ways). However, this overcounts tickets where we match all three white balls (counted three times for each red ball).
Correcting the overcount: There are 220 ways to pick three white balls and only 1 way to pick the red ball (doesn't matter which) for a total of 220 tickets where we match all three white balls. Subtracting this from the overcount gives us the actual number of winning tickets with at least two white ball matches.
Matching the red SuperBall:
Favorable outcomes: Here, we can pick any three white balls (220 ways) as long as they don't match the chosen red SuperBall (8 ways).
Calculations:
Matching at least two white balls:
Total outcomes (overcounted): 220 (white balls) * 8 (red balls) = 1760
Overcounted matches (all three white balls): 220 (white balls) * 1 (red ball) = 220
Corrected favorable outcomes: 1760 (total) - 220 (overcounted) = 1540
Matching the red SuperBall:
Favorable outcomes: 220 (white balls) * 8 (red balls that don't match) = 1760
Total Probability:
We win if either of these scenarios occurs. So, the total probability is the sum of the probabilities of each scenario:
Probability (matching at least two white balls): 1540 favorable outcomes / (12C3 * 8) total outcomes = 1540 / (220 * 8) = 44 / 248
Probability (matching the red SuperBall): 1760 favorable outcomes / (12C3 * 8) total outcomes = 1760 / (220 * 8) = 11 / 31
Total Probability (winning the SuperLottery):
(44 / 248) + (11 / 31) = 33/62.
Therefore, the probability that you win a super prize is 33/62.
