Question

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 one purple one. She grabs eight of them. Find the probability of the following event, expressing it as a fraction in lowest terms. She does not have all the red ones.

Answer 1

0.8586 = 85.86% probability that she does not have all the red ones.

Explanation:

The marbles are grabbed without replacement, which means that the hypergeometric distribution is used to solve this question.

Hypergeometric distribution:

The probability of x successes is given by the following formula:

`P(X = x) = h(x,N,n,k) = (C_(k,x)*C_(N-k,n-x))/(C_(N,n))`

In which:

x is the number of successes.

N is the size of the population.

n is the size of the sample.

k is the total number of desired outcomes.

In this question:

Bag has 4 + 4 + 3 + 1 = 12 marbles, which means that
`N = 12`

She grabs eight of them, which means that
`n = 8`

Four are red, which means that
`k = 4`

Probability that she does not have all the red ones.

This is

`P(X < 4) = 1 - P(X = 4)`

In which

`P(X = x) = h(x,N,n,k) = (C_(k,x)*C_(N-k,n-x))/(C_(N,n))`

`P(X = 4) = h(4,12,8,4) = (C_(4,4)*C_(8,0))/(C_(12,8)) = 0.1414`

Then

`P(X < 4) = 1 - P(X = 4) = 1 - 0.1414 = 0.8586`

0.8586 = 85.86% probability that she does not have all the red ones.