Clarify the problem and define the random variable for the payout. Use linearity of expectation by conditioning on the maximum value M, then compute E[payout] = sum_{m=1}^6 m * P(max = m and results differ). Alternatively, enumerate all 36 equally likely outcomes and average the payouts.
Pro tip: After computing the expected value, mention that you can verify the result by writing a quick simulation or by enumerating all outcomes, showing you value correctness and testing.
Confirm that the die is fair, the two rolls are independent, and the payout is the higher value only if the two results are different. If they match, payout is $0.
Let X be the payout. We need E[X]. Consider conditioning on the maximum of the two rolls, M = max(R1, R2).
For each m from 1 to 6, find P(M = m and R1 ≠ R2). This is the number of ordered pairs (R1, R2) with max = m and R1 ≠ R2, divided by 36.
Compute E[X] = sum_{m=1}^6 m * P(M = m and R1 ≠ R2). Simplify the sum to get the final answer.
Check the result by enumerating all 36 outcomes or by simulation. Discuss any assumptions and potential edge cases.
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