Most Probable Sum by rolling $2$ dice in $[100,111]$.

Question

Given two dices, we roll them and add the result to a sum (initialised to 0) till sum is $\ge 100.$ The resultant sum can be any number in [100 111]. Which among them have the highest probability of being the resultant sum.

Answer 1

The probabilities for the possible final sums are: $$ \begin{array}{cc} 100 & 0.1428571427 \\ 101 & \textbf{0.1428571428} \\ 102 & 0.1388888889 \\ 103 & 0.1309523810 \\ 104 & 0.1190476191 \\ 105 & 0.1031746032 \\ 106 & 0.08333333337 \\ 107 & 0.05952380953 \\ 108 & 0.03968253967 \\ 109 & 0.02380952379 \\ 110 & 0.01190476189 \\ 111 & 0.003968253964 \\ \end{array} $$ Thus, 101 has the highest probability by a very small margin.

So how is this computed? Let $a_k$ be the probability that the number $k$ appears as a sum at some point in the game. Then $a_0=1, a_k=0$ for $k<0,$ and $$ a_k = \frac{1}{36} \sum_{i=1}^6 \sum_{j=1}^6 a_{k-i-j} \cdot \chi_{n-i-j<100}, $$ where $\chi_s$ is 1 if $s$ is true, and 0 otherwise.

I have computed this using Mathematica:

a[0] := 1

a[n_] := 0 /; n < 0

a[n_] := a[n] = Sum[If[n - i - j < 100, a[n - i - j], 0], {i, 1, 6}, {j, 1, 6}]/36

Also, the expected sum (this has not been asked) is about 103.41666666745046042. This can be computed from a similar recursion.

Answer 2

The single die version of this question is analyzed here: Probability of dice sum just greater than 100

In my paper Linear recurrences via probability, American Mathematical Monthly 122, 386-389 (April 2015), I consider the general version of this game. In particular, equation (5) gives the asymptotic hitting probabilities as the lower limit (100 in this problem) gets large.

For the 2 dice game, the asymptotics say that $$\mathbb{P}(\mbox{games ends at state }99+k)\approx{\mathbb{P}(\xi\geq k)\over\mathbb{E}(\xi)}, \mbox{ for }k=1,2,\dots, 12,$$ where $\xi$ is the sum on a pair of dice. These probabilities are $$1/7, 1/7, 5/36, 11/84, 5/42, 13/126, 1/12, 5/84, 5/126, 1/42, 1/84, 1/252,$$ or in decimal form $$.1429, .1429, .1389, .1310, .1190, .1032, .08333, .05952, .03968, .02381, .01190, .003968.$$