Find the Most Likely Stopping Sum When Rolling a Die

# Find the Most Likely Stopping Sum When Rolling a Die Roll a fair six-sided die repeatedly and add the outcomes. Stop as soon as the running sum is at least 61. Among the possible final sums 61, 62, 63, 64, 65, and 66, which is most likely? Give a rigorous argument rather than relying only on simulation. ### What a Strong Answer Covers - The possible predecessor sums immediately before the final roll. - A probability expression using the chance of ever reaching each predecessor. - A comparison that proves which terminal sum is most likely without assuming a uniform overshoot. ```hint Compare adjacent final sums Write the last-step probability for 61 and for 62; see which predecessor term is present in only one expression. ``` ### Follow-up Questions - How does the argument extend to another positive threshold? - Why are the possible overshoots not equally likely?