Carmen is going to roll an 8-sided die 200 times. She predicts that she will roll a multiple of 4 twenty-five times. Based on the theoretical probability, which best describes Carmen's prediction?
A. Carmen's prediction is exact because [tex]$200 \times \frac{1}{8}$[/tex] is 25.
B. Carmen's prediction is low because [tex]$200 \times \frac{1}{4}$[/tex] is 50.
C. Carmen's prediction is low because [tex]$200 \div 2$[/tex] is 100.
D. Carmen's prediction is high because [tex]$200 \div 25$[/tex] is 8.
Answer (2)
Determine the multiples of 4 on an 8-sided die: 4 and 8.
Calculate the probability of rolling a multiple of 4: 8 2 = 4 1 .
Calculate the expected number of times a multiple of 4 will be rolled in 200 rolls: 200 × 4 1 = 50 .
Carmen's prediction is low because the expected number of rolls is 50, and her prediction is 25. $\boxed{Carmen's prediction is low because 200 × 4 1 is 50}$.
Explanation
Analyze the problem First, let's analyze the problem. Carmen is rolling an 8-sided die 200 times and predicts she'll roll a multiple of 4 twenty-five times. We need to determine if her prediction is accurate based on theoretical probability.
Identify multiples of 4 To find the theoretical probability of rolling a multiple of 4 on an 8-sided die, we need to identify the multiples of 4 within the numbers 1 to 8. The multiples of 4 are 4 and 8. So, there are 2 favorable outcomes.
Calculate the probability The probability of rolling a multiple of 4 is the number of favorable outcomes divided by the total number of possible outcomes. In this case, it's 8 2 , which simplifies to 4 1 .
Calculate the expected number of rolls Now, we need to calculate the expected number of times Carmen will roll a multiple of 4 in 200 rolls. We multiply the probability of rolling a multiple of 4 by the number of rolls: 200 × 4 1 = 50 .
Compare prediction to expected rolls Carmen predicts she will roll a multiple of 4 twenty-five times, but based on the theoretical probability, we expect her to roll a multiple of 4 fifty times. Since 25 is less than 50, Carmen's prediction is low.
Conclusion Therefore, the best description of Carmen's prediction is that it is low because 200 × 4 1 is 50.
Examples
Understanding probability helps in many real-life scenarios. For example, if you're playing a game of chance, knowing the probabilities can help you make informed decisions. Similarly, in business, understanding probabilities can help you assess risks and make better investments. In this case, calculating the expected number of times an event occurs helps in making predictions.
Carmen's prediction that she will roll a multiple of 4 twenty-five times is low based on theoretical probability. The expected number of rolls for multiples of 4 in 200 rolls is 50. Therefore, option B is the correct answer: Carmen's prediction is low because 200 × 4 1 is 50.
;
