## PROBLEM 20. ROLLING DICE

A regular die has the numbers $1,2,3,4,5,6$ on its faces. It is rolled randomly until the sum of the points that come up exceeds the number 12.

What is the most likely total sum of points?
Solution. Consider the penultimate roll. After it, the total sum should be either $12,11,10,9,8$,
or 7. If it is 12, the overall result will with equal probability take the values $13,14,15$, $16,17,18$. Similarly, with a sum of 11, the final result will with equal probability take the values 13, 14, $15,16,17$, and so on. The number 13 appears as an equal candidate in each case and is the only number of this kind. Therefore, the number 13 is the most probable.

In general, the same arguments show that the most probable sum, first exceeding $n(n \geqslant 6)$, is $n+1$.