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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?

 Jan 28, 2023
 #1
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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

 Jan 28, 2023
 #2
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1 - [(3/12) (2/11) (1/10)] = 1 / 220

 

2 - [+ 3 (3/12) (2/11) (9/10)] = 27 / 220

 

3 - [+ (1/8 - (1/8) (3/12) (2/11) (1/10))] = 219 / 1760

 

4 - [- (1/8) 3 (3/12) (2/11) (9/10)] = - 27 / 1760

 

5 - The probability is: [1 / 220  +  27 / 220  +  219 / 1760]  -  27 / 1760 = 13 / 55

 Jan 28, 2023
 #3
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The probability of winning a super prize is the sum of the probabilities of matching at least two white balls and the probability of matching the red SuperBall.

 

The probability of matching at least two white balls is 1 minus the probability of not matching any white balls. To match none of the white balls, we first choose three distinct numbers from 1 to 12 that do not match the three white balls that are drawn. This can be done in 12 choose 3 ways (or 220 ways) for the white balls, and then in 8 choose 1 ways for the red SuperBall, for a total of 220*8 = 1760 ways. Since there are 12 choose 3 ways to choose 3 numbers from 1 to 12, the probability of not matching any white balls is 1760/220 = 8/11. Therefore, the probability of matching at least two white balls is 1 - 8/11 = 3/11.

 

The probability of matching the red SuperBall is 1/8, since there are 8 red SuperBalls out of a total of 20.

Therefore, the probability of winning a super prize is (3/11) + (1/8) = 29/44.

 Jan 29, 2023

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