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) =

Explanation:

1. P(alto first) ⋅ P(soprano second)

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

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

Thus:

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

what are the statments

step-by-step explanation:

required rate of return = 0.166 = 16.6%

step-by-step explanation:

under the capital asset pricing model (capm), the required or expected rate of return for a particular security or asset is given using the formula:

required rate of return = risk-free rate + beta*(market risk premium)

we are given that;

risk-free rate = 4% = 0.04

beta = 1.4

market risk premium = 9% = 0.09

we simply plug these values into the above formula;

required rate of return = 0.04 + 1.4*(0.09)

required rate of return = 0.04 + 0.126

required rate of return = 0.166 = 16.6%