Pseudo-dice probability distribution Given the following data, is there an algorithmic way to generate a probability distribution?
Data: min, max, and average results of "die".
Known Trend: results will always be weighted towards lower values, as the gulf between the average and max is greater than the gulf between the average and min (e.g. min=10, avg=19.4 max=42)
Constraints:

*

*Results are restricted to integer values

*The probabilities of any two results should be as close to equal as possible.

Allowances: Rough approximations (+/-0.5) of a given average are acceptable.

First, find $p_1,\ldots,p_{33}$ that satisfy the system
$$
\begin{cases}
p_1+p_2\cdots+p_{33}=1\\
10p_1+11p_2+\cdots+42p_{33}=19.4\cdot33,
\end{cases}
$$
s.t. $0\leq p_i\leq 1$ for all $i=1,\ldots,33$. Then sample $\{10,11,\ldots,42\}$ with cell probabilities $(p_1,p_2,\ldots,p_2)$.
