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It's known that it's impossible to assign probabilities to a pair of loaded dice so that the sums $2,...,12$ are equally probable. How would one set the probabilities $\{p_i: 1\le i\le 6\}$ and $\{q_i: 1\le i\le 6\}$ for the two dice so that $\sum_{i=1}^{11}\|s_i-1/11\|_2$ is minimal? ( for $1\le i\le 11$, $s_i$ is the probability that the sum is $i+1$).

Katy
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1 Answers1

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Let the dice be $A$ and $B$, and the probabilities of each numbered face be $\{a_i,b_i: 1\le i\le 6\}$.

Then the probability of each sum is:

$$\begin{matrix} 2&a_1b_1\\ 3&a_1b_2+a_2b_2\\ 4&a_1b_3+a_2b_2+a_3b_1\\ 5&a_1b_4+a_2b_3+a_3b_2+a_4b_1\\ 6&a_1b_5+a_2b_4+a_3b_3+a_4b_2+a_5b_1\\ 7&a_1b_6+a_2b_5+a_3b_4+a_4b_3+a_5b_2+a_1b_6\\ 8&a_2b_6+a_3b_5+a_4b_4+a_5b_3+a_6b_2\\ 9&a_3b_6+a_4b_5+a_5b_4+a_6b_3\\ 10&a_4b_6+a_5b_5+a_6b_4\\ 11&a_5b_6+a_6b_5\\ 12&a_6b_6\\ \end{matrix}$$

This is a linear programming problem to minimize the least squares error subject to the constraint that all probabilities must be positive. This is a numerical problem that lots of software can do, including the Solver add in of excel.

One solution is

A B

0.243883042 0.243883042

0.137478805 0.137478805

0.118637979 0.118637979

0.118638094 0.118638094

0.137479191 0.137479191

0.243882889 0.243882889

which has a least squares error of $.013$ and chances of:

2 0.059478938

3 0.067057498

4 0.076768004

5 0.090488054

6 0.113753104

7 0.184908719

8 0.11375296

9 0.090488084

10 0.07676813

11 0.067057644

12 0.059478863

Dale M
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  • With different starting values, Excel finds the following better solution. Die A probabilities (rounded) $(.50,0,0,0,0,.50)$, die B probabilities (rounded) $(.125,.1875,.1875,.1875,.1875,.125)$. The least squares error is $0.00284$, and the chances are $(0.625, .09375, .09374, .09376, .09375, .125, .09375, .09374, .09375, .093475, .0625)$. – Steve Kass Oct 15 '14 at 04:53
  • Even better! Are there more ... – Dale M Oct 15 '14 at 06:04
  • I tried a few other choices of initial value without finding anything else. Sometimes your solution appeared, and sometimes mine. The "GRG" engine I used guarantees a local minimum, and while Excel's Simplex LR method guarantees a global solution, I didn’t see how to set up the question to meet its requirements. – Steve Kass Oct 15 '14 at 12:15
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    One can show by analytical computation that Dale's dice weights are algebraic, and can be written as rational linear combinations of algebraic numbers of degree $\leq 18$ over $\mathbb{Q}$. – Travis Willse Oct 15 '14 at 12:49