To estimate the probability that the next three songs she plays are all country songs, we need to design a simulation that accurately represents the equal likelihood of choosing between rock and country songs.
A good simulation would involve an equal split between the two options (rock and country). Let's evaluate the given options to find the best fit.
Option A:
- Number cube (six faces).
- Even number = rock.
- Odd number = country.
- Roll the cube three times. Repeat.
Since a standard six-sided number cube has three even numbers (2, 4, 6) and three odd numbers (1, 3, 5), this approach results in an equal probability (50%) of selecting rock or country for each roll. Consistently rolling the cube three times and categorizing the results correctly achieves the goal. This is a valid simulation.
Option B:
- Random digits (0–9).
- Let [tex]$1,2,3,4,5=$[/tex] rock.
- Let [tex]$6,7,8,9=$[/tex] country.
- Select three random digits. Repeat.
Here, the probability of selecting a rock song is [tex]\(\frac{5}{10}\)[/tex] (or 50%), and the probability of selecting a country song is [tex]\(\frac{4}{10}\)[/tex] (or 40%). This design does not provide an equal likelihood for rock and country songs. This is not a valid simulation.
Option C:
- Number cube (which typically has six faces).
- Let [tex]$1=$[/tex] rock.
- Let [tex]$2=$[/tex] country.
- Roll cube three times. Repeat.
This option fails because it only specifies outcomes for two results (rock for [tex]$1$[/tex] and country for [tex]$2$[/tex]) on the number cube without consideration for the remaining four numbers ([tex]$3, 4, 5, 6$[/tex]). This is not a valid simulation.
Option D:
- Random digits.
- Let [tex]$1,2,3=$[/tex] rock.
- Let [tex]$4,5,6=$[/tex] country.
- Select three random digits. Repeat.
Here, the probability of selecting a rock song is [tex]\(\frac{3}{6}\)[/tex] (or 50%), and the probability of selecting a country song is also [tex]\(\frac{3}{6}\)[/tex] (or 50%). This approach means there is an equal chance of selecting either genre, matching the need for equal likelihood. This is a valid simulation.
Conclusion:
Both Option A and Option D provide valid ways to estimate the probability of playing three consecutive country songs because they both ensure an equal likelihood of selecting rock or country songs.
To summarize:
- Simulation Design A: Valid.
- Simulation Design D: Valid.
In the context of the question, either Option A or Option D can be used to accurately estimate the probability.
