Question
Read the story.
Alice likes to play a trivia game on her phone. She is able to answer the question correctly 75% of the time. How likely is it that she will answer 5 questions correctly in a row?
Which simulation could be used to fairly represent the situation?
Alice likes to play a trivia game on her phone. She is able to answer the question correctly 75% of the time. How likely is it that she will answer 5 questions correctly in a row?
Which simulation could be used to fairly represent the situation?
Answers
GPT-4o mini
To simulate the situation where Alice answers trivia questions with a 75% success rate, you could employ a simple Bernoulli trial simulation. Here's how you can represent her answering questions using a simulation:
1. **Define Success**: Since Alice answers correctly 75% of the time, you can consider a "success" as her answering a question correctly.
2. **Define Failure**: A "failure" would be her answering incorrectly, which has a probability of 25%.
3. **Random Number Generation**: You can use a random number generator to simulate the results of her answers. For instance, you could generate a number between 1 and 100:
- If the number is between 1 and 75 (inclusive), it represents a correct answer.
- If the number is between 76 and 100, it represents an incorrect answer.
4. **Run the Simulation**: You would repeat the random number generation for 5 questions. You need to check whether all 5 random numbers fall within the range of success (1 to 75) to see if she answered all 5 questions correctly.
5. **Repeat for Multiple Trials**: Run the entire simulation multiple times (e.g., 1,000 or more trials) to estimate the frequency of how often she gets all 5 questions correct in a row.
Calculating the probability without simulation:
The probability of Alice answering 5 questions correctly in a row is \(0.75^5\):
\[
P(5 \text{ correct}) = 0.75^5 = 0.2373 \text{ (approximately)}
\]
This means that based on pure probability, Alice has about a 23.73% chance of answering 5 questions correctly in a row.
Using this simulation method will allow for a practical understanding of her success over many trials in a manner akin to real-world situations.
1. **Define Success**: Since Alice answers correctly 75% of the time, you can consider a "success" as her answering a question correctly.
2. **Define Failure**: A "failure" would be her answering incorrectly, which has a probability of 25%.
3. **Random Number Generation**: You can use a random number generator to simulate the results of her answers. For instance, you could generate a number between 1 and 100:
- If the number is between 1 and 75 (inclusive), it represents a correct answer.
- If the number is between 76 and 100, it represents an incorrect answer.
4. **Run the Simulation**: You would repeat the random number generation for 5 questions. You need to check whether all 5 random numbers fall within the range of success (1 to 75) to see if she answered all 5 questions correctly.
5. **Repeat for Multiple Trials**: Run the entire simulation multiple times (e.g., 1,000 or more trials) to estimate the frequency of how often she gets all 5 questions correct in a row.
Calculating the probability without simulation:
The probability of Alice answering 5 questions correctly in a row is \(0.75^5\):
\[
P(5 \text{ correct}) = 0.75^5 = 0.2373 \text{ (approximately)}
\]
This means that based on pure probability, Alice has about a 23.73% chance of answering 5 questions correctly in a row.
Using this simulation method will allow for a practical understanding of her success over many trials in a manner akin to real-world situations.