A group of nine singers includes 6 altos and 3 sopranos. An instructor randomly selects one of the singers to sing the first part of a duet and a different singer to sing the second part of a duet. Use a sample space to determine whether randomly selecting an alto first and randomly selecting a soprano second are independent events.

P (alto first) ⋅ P (soprano second) =

P (alto first and soprano second) =

Yes, they are independent events, as proved below.

Explanation:

1. P (alto first) ⋅ P (soprano second)

P (alto first) = number of altos / number of singers = 6/9 = 2/3

P (soprano second) = number of sopranos / number of singers remaining = 3/8

P (alto first) ⋅ P (soprano second) = 2/3 * 3/8 = 2/8 = 1/4

2. P (alto first and soprano second)

The number of events or elements in the set of "alto first and soprano second" is equal to the number of elements contained in the set of altos multiplied by the number of elements contained in the set of sopranos.

That is

6 * 3 = 18elements.

The sample space for selecting two singers from the nine singers is:

9 for the first selection * 8 for the second selection = 72 combinations.

Thus:

P (alto first and soprano second) = 18/72 = 1/4

Therefore, by definition, since P (alto first) ⋅ P (soprano second) = P (alto first and soprano second), the events are independent.