Most likely final die face when passing 1000

Question

A fair die is rolled repeatedly and the faces are added up, stopping as soon as the total is $1000$ or more. Which face is most likely to be the one rolled last?

Short answer

$6$ . Far from the start, the running total hits any given value with probability close to $\tfrac1{3.5}$ . A final face $k$ can come from any of the $k$ totals $1000-k,\dots,999$ , so $P(\text{last}=k)$ is roughly proportional to $k$ and $6$ is the most likely (about $\tfrac{6}{21}$ ).

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Counting and Combinatorics
Expectation, Mean, and Variance
Series and the Geometric Series: Partial Sums and Convergence

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