Question

a jar contains ten black buttons and six brown buttons. If nine buttons are picked at random, what is the probability that exactly five of them are black?

Answer 1

33.04% probability that exactly five of them are black

Explanation:

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

The order in which the buttons are chosen 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)!)`

Desired outcomes:

Five black buttons, from a set of 10

Four brown buttons, from a set of 6.

`D = C_(10,5)*C_(6,4) = (10!)/(5!(10-5)!)*(6!)/(4!(6-4)!) = 3780`

Total outcomes:

Nine buttons, from a set of 16.

`T = C_(16,9) = (16!)/(9!(16-9)!) = 11440`

Probability:

`p = (D)/(T) = (3780)/(11440) = 0.3304`

33.04% probability that exactly five of them are black