Answer :
To estimate the probability that the next three songs played are all country songs, we'll need to set up a simulation that accurately reflects the conditions given – i.e., an equal number of rock, country, and pop songs, and randomly selecting three songs.
### Simulation Design
We need to simulate the process of selecting three songs out of three categories (rock, country, pop) and see how often all three selected songs are country songs.
#### Option A: Coin
This option uses a coin toss:
- Heads (H) represents a country song.
- Tails (T) represents either rock or pop songs.
Procedure:
1. Toss a coin three times.
2. Count the number of times that the outcomes are HHH (all heads), which would correspond to playing three country songs.
3. Repeat this experiment multiple times (e.g., 1000 times).
4. The probability estimate is the number of HHH outcomes divided by the total number of trials.
This method does not differentiate between rock and pop specifically, but it simplifies the problem since tails (T) can represent either rock or pop.
#### Option B: Random Digits
This option uses random digits:
- Let digits 1, 2, 3 represent rock songs.
- Let digits 4, 5, 6 represent country songs.
- Let digits 7, 8, 9 represent pop songs.
- Digit 0 is not used or can be treated in any chosen consistent way (for simplicity).
Procedure:
1. Randomly select a digit three times.
2. Check if all three selected digits are in the range 4, 5, 6; if they are, it represents three country songs.
3. Repeat this experiment multiple times (e.g., 1000 times).
4. The probability estimate is the number of times all three selected digits are in the range 4-6 divided by the total number of trials.
### Evaluating the Options
Both methods could be used to simulate the probability, but option B provides a more direct and clear representation of the three equal categories (rock, country, pop).
### Conclusion
Option B (Random Digits) is the better simulation design because it clearly distinguishes between rock, country, and pop songs by using specific ranges of digits. This method more accurately reflects the problem's conditions, ensuring that the probability estimation is precise. Therefore, to estimate the probability that the next three songs are all country songs, the DJ should use:
B. Random digits
Let [tex]$1,2,3=$[/tex] rock
Let [tex]$4,5,6=$[/tex] country
Let [tex]$7,8,9=$[/tex] pop
### Simulation Design
We need to simulate the process of selecting three songs out of three categories (rock, country, pop) and see how often all three selected songs are country songs.
#### Option A: Coin
This option uses a coin toss:
- Heads (H) represents a country song.
- Tails (T) represents either rock or pop songs.
Procedure:
1. Toss a coin three times.
2. Count the number of times that the outcomes are HHH (all heads), which would correspond to playing three country songs.
3. Repeat this experiment multiple times (e.g., 1000 times).
4. The probability estimate is the number of HHH outcomes divided by the total number of trials.
This method does not differentiate between rock and pop specifically, but it simplifies the problem since tails (T) can represent either rock or pop.
#### Option B: Random Digits
This option uses random digits:
- Let digits 1, 2, 3 represent rock songs.
- Let digits 4, 5, 6 represent country songs.
- Let digits 7, 8, 9 represent pop songs.
- Digit 0 is not used or can be treated in any chosen consistent way (for simplicity).
Procedure:
1. Randomly select a digit three times.
2. Check if all three selected digits are in the range 4, 5, 6; if they are, it represents three country songs.
3. Repeat this experiment multiple times (e.g., 1000 times).
4. The probability estimate is the number of times all three selected digits are in the range 4-6 divided by the total number of trials.
### Evaluating the Options
Both methods could be used to simulate the probability, but option B provides a more direct and clear representation of the three equal categories (rock, country, pop).
### Conclusion
Option B (Random Digits) is the better simulation design because it clearly distinguishes between rock, country, and pop songs by using specific ranges of digits. This method more accurately reflects the problem's conditions, ensuring that the probability estimation is precise. Therefore, to estimate the probability that the next three songs are all country songs, the DJ should use:
B. Random digits
Let [tex]$1,2,3=$[/tex] rock
Let [tex]$4,5,6=$[/tex] country
Let [tex]$7,8,9=$[/tex] pop
