Question

Sonya has two red marbles and three yellow marbles. She chooses three marbles at random. What is the probability that she has at least one marble of each color?

Answer 1

90% probability that she has at least one marble of each color

Explanation:

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

The order in which the marbles are selected is not important. So we use the combinations formula to solve this question.

Combinations formula:

`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

What is the probability that she has at least one marble of each color?

Desired outcomes:

Two red(from a set of 2) and one yellow(from a set of 3)

Or

One red(from a set of 2) and two yellows(from a set of 3).

So

`D = C_(2,2)*C_(3,1) + C_(2,1)*C_(3,2) = (2!)/(2!(2-2)!)*(3!)/(1!(3-1)!) + (2!)/(1!(2-1)!)*(3!)/(2!(3-2)!) = 3 + 6 = 9`

Total outcomes:

Three marbles, from a set of 3 + 2 = 5. So

`T = C_(5,3) = (5!)/(3!(5-3)!) = 10`

Probability:

`p = (D)/(T) = (9)/(10) = 0.9`

90% probability that she has at least one marble of each color