Question

Use the counting techniques from the last chapter. A bag contains three red marbles, three green ones, one fluorescent pink one, two yellow ones, and four orange ones. Suzan grabs four at random. Find the probability of the indicated event.
She gets at least two red ones, given that she gets at least one green one.

Answer 1

Using the combination formula and the probability concept, it is found that there is a 0.1259 = 12.59% probability that she gets at least two red ones, given that she gets at least one green one.

• A probability is the number of desired outcomes divided by the number of total outcomes.

• In this problem, the order in which the marbles are taken is not important, hence, the combination formula is used to solve this question.

Combination formula:

$C_{n,x}$ is the number of different combinations of x objects from a set of n elements, given by:

$C_{n,x} = \frac{n!}{x!(n-x)!}$

Desired outcomes:

• 2 red from a set of 3.

• 1 green from set of 3.

• 1 from a set of 1 + 2 + 1 + 2 + 4 = 10.

Hence:

$D = C_{3,2}C_{3,1}{C_{10,1}} = \frac{3!}{2!1!} \times \frac{3!}{1!2!} \times \frac{10!}{1!9!} = 90$

Total outcomes:

Four marbles are taken from a set of 13, hence:

$T = C_{13,4} = \frac{13!}{4!9!} = 715$

Then, the probability is:

$p = \frac{D}{T} = \frac{90}{715} = 0.1259$

0.1259 = 12.59% probability that she gets at least two red ones, given that she gets at least one green one.

To learn more about the use of the combination formula and the probability concept, you can check combination formula and the probability concept,