To estimate the probability that the next three songs to play are all hip-hop songs on Brandi's music player, we need to design a simulation that accurately reflects the scenario where there are equal numbers of hip-hop and country songs. Since the songs are played on a random shuffle, each song has an equal probability of being either hip-hop or country, making the probability of selecting a hip-hop song 1/2 and the probability of selecting a country song 1/2 for each trial.
Let's analyze each simulation design option:
A. Coin
- Let heads (H) = hip-hop
- Let tails (T) = country
- Toss coin three times. Repeat.
This option uses a fair coin to represent the equal probability (1/2) of selecting either a hip-hop or country song. By tossing the coin three times, you simulate the selection of three consecutive songs. This accurately represents the problem conditions of equal probabilities and three trials.
B. Number cube
- Let 1 = hip-hop
- Let 2 = country
- Roll cube four times. Repeat.
This option uses a number cube (die), but it only assigns the outcomes of 1 and 2 to the songs. There's an issue as a number cube typically has six faces, and this setup doesn't account for the remaining faces (3, 4, 5, 6). Additionally, it specifies rolling the cube four times instead of three. Therefore, this design does not accurately reflect the specified conditions.
C. Number cube
- Let 1 = hip-hop
- Let 2 = country
- Roll cube three times. Repeat.
This option is similar to option B but adjusted for three trials. However, it still only assigns outcomes of 1 and 2 to the songs, leaving other outcomes on the number cube undefined. A typical number cube has six faces, thus this setup might cause confusion as it does not utilize all faces appropriately.
D. Random digits
- Let 0,1 = hip-hop
- Let 2,3,4,5,6 = country
This option uses random digits, but the probabilities are not equal. There are only two digits for hip-hop and five for country, which gives hip-hop a probability of 2/7 and country a probability of 5/7. This setup does not represent an equal probability condition as required.
Given the analysis, the best simulation design to estimate the probability that the next three songs are all hip-hop, reflecting equal probability with a 1/2 chance for each, is option A:
A. Coin
Let heads (H) = hip-hop
Let tails (T) = country
Toss coin three times. Repeat.
