Martin obtains a biased five-sided die, with the even numbers having twice the likelihood of appearing compared to the odd numbers. He chooses to simulate six rolls of this biased die, conducting 1,000 replications, with the objective of determining the probability that the highest number rolled in each simulation is equal to 5. Which of the following would produce the probability Martin desires?

Question

BlackTom AI · Accepted Answer

The question asks which approach would yield the probability that the maximum value in six rolls of a biased five-sided die equals 5, given that even numbers are twice as likely as odd numbers.
First, the die’s face values are 1, 2, 3, 4, 5 with probabilities proportional to the pattern 1,2,1,2,1. Normalizing, the probabilities are (1/7, 2/7, 1/7, 2/7, 1/7) for faces 1 through 5 respectively. To estimate P(max of six rolls = 5), we can simulate many replications of six independent rolls with these probabilities, take the maximum in each replication, and then compute the fraction of replications where the maximum equals 5.
The provided code snippet does exactly that in R:
- It uses replicate(1000, …) to run 1000 independent replications.
- Inside, sample(seq_len(5), 6, replace = TRUE, prob = c(1, 2, 1, 2, 1)) draws six rolls from {1,2,3,4,5} with the desired probabilities (the prob vector is unnormalized here, but R uses it directly as weights; effectively it treats them as proportional probabilities).
- max(...) computes the maximum value among the six draws in that replication.
- dice_rolls <- replicate(...) stores the 1000 maxima.
- mean(dice_rolls == 5) computes the proportion of replications where the maximum equals 5, which is the estimated P(max = 5).

If there were alternative options, you’d evaluate them by similar logic:
- Any approach that uses six draws with the same biased distribution and then checks if the maximum is 5 would be valid.
- Approaches that use fewer than six draws, or that switch the target event to something like max < 5 or max ≤ 4, would estimate different probabilities.
- If the probability vector did not reflect the bias (for example, using equal probabilities for all faces), the resulting estimate would not reflect the intended bias.

Note on the absence of answer choices: The provided data shows an answer field containing a code snippet, but the answer_options array is empty. This means there are no multiple-choice options to evaluate here. In a typical MCQ, you would compare each option’s method to this reasoning and select the one that matches the described simulation approach. In this case, the code given (computing the maximum over 6 draws with the specified face probabilities and then estimating the proportion equal to 5) is the correct strategy to obtain the desired probability, assuming the implementation details align with the described probabilities.

PSTAT 10 W25 Quiz 3

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