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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?

Sep 3, 2023

#1
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The number of ways to choose 3 white balls from 12 is 12C3.

12C3 = (12 * 11 * 10) / (3 * 2) = 220

The number of ways to choose 1 red ball from 8 is 8C1.

8C1 = 8

The number of ways to choose 3 numbers from 1 to 12 and 1 number from 13 to 20 is 12C3 * 8.

12C3 * 8 = 220 * 8 = 1760

The number of ways to win a super prize is the number of ways to match at least 2 white balls or the red SuperBall.

There are 3 ways to match 2 white balls:

Choose 2 white balls and the red SuperBall. The number of ways to do this is 12C2 * 8.

12C2 = (12 * 11) / (2 * 1) = 66

Choose 3 white balls. This can be done in 12C3 ways.

Choose the red SuperBall and 1 white ball. The number of ways to do this is 12 * 8.

12 * 8 = 96

So the total number of ways to match 2 white balls is 12C2 * 8 + 12C3 + 12 * 8 = 312.

The number of ways to match the red SuperBall only is 12 * 8 = 96.

Therefore, the total number of ways to win a super prize is 312 + 96 = 408.

The probability of winning a super prize is the number of winning tickets divided by the total number of tickets.

Probability of winning = 408 / 1760 = 51/220.

Sep 3, 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

Sep 3, 2023
#3
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To find the probability of winning a super prize in the SuperLottery, we need to consider two scenarios:

Matching at least two of the white balls.

Matching the red SuperBall.

We'll calculate the probability for each scenario separately and then add them together.

Scenario 1: Matching at least two of the white balls

There are a total of 12 white balls, and we need to choose 3 of them for our ticket. The total number of ways to choose 3 white balls out of 12 is given by the binomial coefficient:

C(12, 3) = 12! / (3!(12-3)!) = 220 ways

Now, for each of these combinations, there are 8 white balls left, and we have to ensure that the fourth ball (the SuperBall) is not one of the 8 remaining white balls. So, for each of the 220 combinations of white balls, there are 8 choices for the SuperBall.

So, the total number of ways to win by matching at least two white balls is:

220 (combinations of white balls) * 8 (choices for the SuperBall) = 1,760 ways

Scenario 2: Matching the red SuperBall

There are 8 red SuperBalls (numbers from 13 to 20), and we need to choose 1 of them for our ticket. So, there are 8 ways to choose the SuperBall.

Now, let's calculate the total number of possible outcomes when drawing a ticket:

For the white balls, there are 12 choices for the first ball, 11 choices for the second ball (since it's drawn without replacement), and 10 choices for the third ball.

For the SuperBall, there are 8 choices.

So, the total number of possible outcomes is:

12 * 11 * 10 * 8 = 10,560

Now, we can calculate the probability of winning a super prize by either matching at least two white balls or matching the SuperBall:

Probability of winning = (Ways to win by matching at least two white balls + Ways to win by matching the SuperBall) / Total possible outcomes

Probability of winning = (1,760 + 8) / 10,560

Probability of winning = 1,768 / 10,560

Now, simplify the fraction:

Probability of winning = 221 / 1320

Sep 3, 2023
#4
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The probability that you win a super prize is 0.2361111111111111.

Here is the calculation:

The number of ways to choose 3 white balls from 12 is 12C3.

Python

number_of_ways_to_choose_white_balls = 12 ** 3

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The number of ways to choose 1 red ball from 8 is 8C1.

Python

number_of_ways_to_choose_red_ball = 8

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The number of ways to choose 3 numbers from 1 to 12 and 1 number from 13 to 20 is 12C3 * 8.

Python

total_number_of_tickets = 12 ** 3 * 8

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The number of ways to win a super prize is the number of ways to match at least 2 white balls or the red SuperBall.

Python

number_of_winning_tickets = ( 12 ** 2 * 8 + 3 * 12 * 7 * 8 + 12 * 8 ) # 12C2 * 8 + 3 * 12C1 * 7C1 * 8 + 12C1 * 8

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The probability of winning a super prize is the number of winning tickets divided by the total number of tickets.

Python

probability_of_winning = number_of_winning_tickets / total_number_of_tickets

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Python

print(probability_of_winning)