Question

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, 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 red ones and one of each of the other colors.

Answer 1

Solution:

Given: a bag contains:

three red marbles,

four green ones,

three white ones, and

two purple ones.

Total = 3+4+3+2 =12 marbles.

Suzan grabs five of them.

Find:

P( She has two red ones and one of each of the other colors) =...........?

P( 2 Red, 1 Green , 1 White , 1 Purple) = .............?

2 red marbles can be selected from 3 red marbles in ₃C₂ ways

1 Green marble can be selected from 4 Green  marbles in ₄C₁ ways

1 White marble can be selected from 3 White marbles in ₃C₁ ways

1 Purple marble can be selected from 2 Purple marbles in ₂C₁ ways

and

any 5 marbles can be selected from 12 marbles in ₁₂C₅ ways

Thus

P( 2 Red, 1 Green , 1 White , 1 Purple) = ₃C₂ X ₄C₁ X ₃C₁ X ₂C₁ / ₁₂C₅

thus

thus

₃C₁ = 3

and   ₂C₁ = 2

thus

P( 2 Red, 1 Green , 1 White , 1 Purple) = ₃C₂ X ₄C₁ X ₃C₁ X ₂C₁ / ₁₂C₅

P( 2 Red, 1 Green , 1 White , 1 Purple) = 3 X 4 X 3 X 2 / 792

P( 2 Red, 1 Green , 1 White , 1 Purple) = 72 / 792

P( 2 Red, 1 Green , 1 White , 1 Purple) = 1 / 11