53. Dice. A die, on the faces of which the numbers $0,1,2,3,4,5$ are depicted, is rolled until the total sum of the points rolled exceeds 12. What is the most likely value of this sum?
53. Consider the penultimate throw. After this throw, the sum of points can be $12, 11, 10, 9$ or 8. If it is 12, then after the last throw, the final sum of points can, with equal probability, take the values $13, 14, 15, 16$ or 17. Similarly, if this sum is 11, then the final sum can, with equal probability, take the values $13, 14, 15$ or 16, and so on. From this, it is clear that the most probable value of the final sum of points is 13.

[N. J. Fine, A. M. M., 55, 98 (February 1948).]

If instead of 12, we take any $N>3$, then the most probable value of the sum is $N+1^{\star}$.