Question

question 5 of 10
a disc jockey at a school dance has equal numbers of rock and country songs that she randomly selects from. she designs a simulation to estimate the probability that the next three songs she plays are all country songs.
which simulation design could she use to estimate the probability?
a. random digits
let 1, 2, 3, 4, 5 = rock
let 6, 7, 8, 9 = country
select three random digits. repeat.
b. random digits
let 1, 2, 3 = rock
let 4, 5, 6 = country
select three random digits. repeat.
c. number cube
let 1 = rock
let 2 = country
roll cube three times. repeat.
d. number cube
let even number = rock
let odd number = country
roll cube three times. repeat.

Explanation:

Step1: Analyze the probability of rock and country songs

Since there are equal numbers of rock and country songs, the probability of choosing a rock song is \(P(\text{rock})=\frac{1}{2}\), and the probability of choosing a country song is \(P(\text{country})=\frac{1}{2}\)

Step2: Analyze each option

• Option A:

The number of rock - related digits (\(1,2,3,4,5\)) is \(5\), and the number of country - related digits (\(6,7,8,9\)) is \(4\). The probabilities \(P(\text{rock})=\frac{5}{9}\) and \(P(\text{country})=\frac{4}{9}\) are not equal.

• Option B:

The number of rock - related digits (\(1,2,3\)) is \(3\), and the number of country - related digits (\(4,5,6\)) is \(3\). But we have only \(6\) digits considered. If we use all \(10\) digits (\(0 - 9\)), this option is not a proper representation of equal - probability for rock and country (because we ignore \(0,7,8,9\)).

• Option C:

A number cube has \(6\) faces. If we let \(1=\text{rock}\) and \(2 = \text{country}\), the probability of getting rock \(P(\text{rock})=\frac{1}{6}\) and the probability of getting country \(P(\text{country})=\frac{1}{6}\), and we are not using all the possible outcomes of the cube in a way that represents the equal - number situation of rock and country songs.

• Option D:

A number cube has \(6\) faces. The number of even numbers (\(2,4,6\)) is \(3\), and the number of odd numbers (\(1,3,5\)) is \(3\). So \(P(\text{rock})=\frac{3}{6}=\frac{1}{2}\) (since even numbers represent rock) and \(P(\text{country})=\frac{3}{6}=\frac{1}{2}\) (since odd numbers represent country)

Answer:

D. Number cube: Let even number = rock, Let odd number = country, Roll cube three times. Repeat.