Question

1. Recall from Example 1 that whenever Suzan sees a bag of marbles, she grabs a handful at random. She has seen a bag containing three red marbles, five green ones, two white ones, and two purple ones. She grabs eight of them. Find the probability of the following event, expressing it as a fraction in lowest terms. HINT [See Example 1.

She has all the red ones.

2. Recall from Example 1 that whenever Suzan sees a bag of marbles, she grabs a handful at random. She has seen a bag containing four red marbles, five green ones, three white ones, and three purple ones. She grabs eight of them. Find the probability of the following event, expressing it as a fraction in lowest terms. HINT [See Example 1.]

She has at least one green one.

3. Recall from Example 1 that whenever Suzan sees a bag of marbles, she grabs a handful at random. She has seen a bag containing four red marbles, two green ones, five white ones, and three purple ones. She grabs five of them. Find the probability of the following event, expressing it as a fraction in lowest terms. HINT [See Example 1.]

She has two red ones and one of each of the other colors.

4. Recall from Example 1 that whenever Suzan sees a bag of marbles, she grabs a handful at random. She has seen a bag containing four red marbles, four green ones, three white ones, and two purple ones. She grabs five of them. Find the probability of the following event, expressing it as a fraction in lowest terms. HINT [See Example 1.]

She has two green ones and one of each of the other colors.

5. Recall from Example 1 that whenever Suzan sees a bag of marbles, she grabs a handful at random. She has seen a bag containing three red marbles, two green ones, four white ones, and one purple one. She grabs seven of them. Find the probability of the following event, expressing it as a fraction in lowest terms. HINT [See Example 1.]

She does not have all the red ones.

6. Recall from Example 1 that whenever Suzan sees a bag of marbles, she grabs a handful at random. She has seen a bag containing four red marbles, three green ones, two white ones, and two purple ones. She grabs eight of them. Find the probability of the following event, expressing it as a fraction in lowest terms. HINT [See Example 1.]

She does not have all the green ones.

Answer 1

Probability = Favorable Outcomes / Total Outcomes

Please note ⁿC_x = n! / [(n-x)!*x!]

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(1) Total Marbles = 3 Red + 5 Green + 2 White = 10

To Find P(All red)

Favorable Outcomes = 3 Red * 5 Others = ³C₃ * ⁷C₅ = 1 * 21 = 21

Total Outcomes = Choosing 8 out of 10 = ¹⁰C₈ = 45

Therefore P(All Red) = 21/45 = 7 / 15

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(2) Total Marbles = 4 Red + 5 Green + 3 White = 12

To Find P(At least 1 green) = P(1 G * 7 Others) + P(2 G * 6 Others) + P(3 G * 5 Others) + P(4 G * 4 Others) + P(5 G * 3 Others)

1 G * 7 Others = ⁵C₁ * ⁷C₇ = 5 * 1 = 5

2 G * 6 Others = ⁵C₂ * ⁷C₆ = 10 * 7 = 70

3 G * 5 Others = ⁵C₃ * ⁷C₅ = 10 * 21 = 210

4 G * 4 Others = ⁵C₄ * ⁷C₄ = 5 * 35 = 175

5 G * 3 Others = ⁵C₅ * ⁷C₃ = 1 * 35 = 35

Therefore Total Favorable Outcomes = 5 + 70 + 350 + 175 + 21 = 495

Total Outcomes = Choosing 8 out of 12 = ¹²C₈ = 495

Therefore P(All Red) = 495/495 = 1

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(3) Total Marbles = 4 Red + 2 Green + 5 White + 3 Purple= 14

To Find P(2 Reds and 1 each of the others)

Favorable Outcomes = ⁴C₂ * ²C₁ * ⁵C₁ * ³C₁   = 6 * 2 * 5 * 3 = 180

Total Outcomes = Choosing 5 out of 14 = ¹⁴C₅ = 2002

Therefore P(2 Red and 1 of each color) = 180/2002 = 90 / 1001

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(4) Total Marbles = 4 Red + 5 Green + 3 White + 2 Purple = 14

To Find P(2 Greens and 1 each of the others)

Favorable Outcomes = ⁴C₁ * ⁵C₂ * ³C₁ * ²C₁   = 4 * 10 * 3 * 2 = 240

Total Outcomes = Choosing 5 out of 14 = ¹⁴C₅ = 2002

Therefore P(2 Red and 1 of each color) = 240/2002 = 180 / 1001

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(5) Total Marbles = 3 Red + 2 Green + 4 White + 1 Purple = 10

To Find P(Not all red) = 1 - P(All red)

Total Outcomes = Choosing 7 out of 10 = ¹⁰C₇ = 120

All red out of 3 and 4 others from the remaining 7 = ³C₃ * ⁷C₄ = 1 * 35 = 35

Favorable Outcomes = 120 - 35 = 85

Therefore P(Not all Red) = 85/120 = 17 / 24

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(6) Total Marbles = 4 Red + 3 Green + 2 White + 2 Purple = 11

To Find P(Not all Green) = 1 - P(All green)

Total Outcomes = Choosing 8 out of 11 = ¹¹C₈ = 165

All Green out of 3 and 5 others from the remaining 8 = ³C₃ * ⁸C₅ = 1 * 56 = 56

Favorable Outcomes = 165 - 56 = 109

Therefore P(Not all Green) = 109 / 165

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