I have the following problem.
Alice produces dices. Alice knows the exact probabilities of each dice. Bob doesn't know anything regarding the distribution of the dice. Alice knows the exact probabilities of each dice and sends the probabilities to Bob using a $deterministic$ $channel$ $C$.
Assuming that Alice sent vector $p$, Bob gets vector $q$. The channel is corrupted and modifies each probability in the following way for $i$ in $\{1, 2, 3, 4, 5, 6\}$ $q_i > p_i$. So when $\sum_{i=1}^{6} p_i = 1$ the received $q$ has this property $\sum_{i=1}^{6} q_i > 1$.
Bob rolls the dice only once and sees $X$.
The above procedure is repeated $N$ times.(Each time Alice produces a new dice and sends it.Bob rolls the dice.)
Is there any rule Bob can follow (based on previous observations of $q$ and the roll $X$) which allows Bob to know the vector $p$?