In a presumably unfair dice the unknown probabilities of appearance of the individual faces are respectively p1, ..., p6 (none of them are zeros).
Through repeated experiments it has been noted that the probability of rolling a streak of 6 consecutive numbers (i.e. rolling a sequence 1,2,..., 6) is greater than or equal to: $(\prod_{i} p_{i}^{p_i})^{6/\sum_{i} p_i}$
What are the possible values of p1, ..., p6 assuming they add up to 1?
What I tried? I tried plugging in values for $i=[1, 6]$ to get 5 inequalities which are really complex to solve for. For eg, $i=2$, $p_2^6 \ge (p_1^{p_1})^{6/p_1}. (p_2^{p_2})^{6/(p_1+p_2)}$