Question

a disc jockey at a school dance has equal numbers of rock, country, and pop 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. coin let heads (h) = country let tails (t) = rock or pop toss coin three times. repeat. b. random digits let 1, 2, 3 = rock let 4, 5, 6 = country let 7, 8, 9, 0 = pop select three random digits. repeat. c. number cube let 1, 2 = rock let 3, 4 = country let 5, 6 = pop roll cube three times. repeat. d. number cube let even number = country let odd number = not country roll cube three times. repeat.

Explanation:

The problem is about simulating a probability where there are three equal - numbered categories (rock, country, pop). We need a simulation where each category has an equal probability of being selected.

• Option A: A coin has 2 outcomes. Here, country is 1 outcome and rock or pop is 1 outcome. But we have three categories, so this doesn't give equal probability for each of the three song types.
• Option B: The number of digits assigned to rock (3 digits: 1,2,3), country (3 digits: 4,5,6), and pop (4 digits: 7,8,9,0) are not equal. So the probabilities of selecting each song type will not be equal.
• Option C: A number cube has 6 outcomes. Rock is assigned 2 outcomes (1,2), country is assigned 2 outcomes (3,4), and pop is assigned 2 outcomes (5,6). So each of the three song types has an equal probability of being selected (since $\frac{2}{6}=\frac{1}{3}$ for each type). When we roll the cube three times, we are simulating the selection of three songs, which is what we need to estimate the probability that all three are country.
• Option D: A number cube has 3 odd and 3 even numbers. This groups the songs into two categories (country and not - country), not three, so it can't be used to simulate the selection of three song types with equal numbers.

Answer:

C. Number cube
Let 1, 2 = rock
Let 3, 4 = country
Let 5, 6 = pop
Roll cube three times. Repeat.