Question

probability that you will roll an even exactly 5 times when you roll a six - sided number cube 10 times. roll a six - sided number cube 20 times. explain why the second result is less. sample response: as the number of trials changes, the distribution changes. since there is a 50% chance of rolling an even number, having a small number of successes is less likely when the number of rolls increases. which of the following ideas did you include in your response? check all of the boxes that apply. the number of trials affects the distribution. 5 successes is the expected value for 10 trials. 5 successes is close to an end of the distribution for 20 trials.

Explanation:

Step1: Analyze the problem

We are considering the probability of rolling an even - numbered result in a number of trials of rolling a number cube.

Step2: Understand the concept of trials and distribution

The number of trials affects the distribution of the results. In a fair number cube, the probability of rolling an even number is 0.5.

Step3: Evaluate each option

• The number of trials affects the distribution: This is correct as stated in the sample response. As the number of trials changes, the distribution of the number of successes (rolling an even number) changes.
• 5 successes is the expected value for 10 trials: The expected value $E(X)=np$, where $n = 10$ and $p=0.5$, so $E(X)=10\times0.5 = 5$. But this is not the main idea in the sample response.
• 5 successes is close to an end of the distribution for 20 trials: For $n = 20$ and $p = 0.5$, the expected value $E(X)=20\times0.5=10$. So 5 successes is on the lower - end of the distribution for 20 trials, which is in line with the sample response that having a small number of successes is less likely when the number of rolls increases.

Answer:

The number of trials affects the distribution, 5 successes is close to an end of the distribution for 20 trials.