Expected number of dice rolls of an unfair dice to roll every side equally many sides

Question

I am having trouble with solving the following problem:

The probability that a $d$-sided dice lands on its $k$th side is equal to $p_k$ for $k\in \{k\in\mathbb{N},k≤d\}$ and $p_1+p_2+p_3+...+p_d=1$. Roll this dice (at least once) until every side is rolled equally many times. Find a function $F(p_1,p_2...)$ which gives the expected number of rolls $n$ after which that happens.

I have attempted to solve it by counting distinct sequences, and than taking a weighted average: the sum of all $n$s times the probability of this occurring after $n$ rolls. Using this method, I have managed to obtain a result for a two-sided dice (a coin) by using Catalan numbers. This does, however, not work for higher dimensions.

Does anyone have an idea on how this could be solved for at least a 3 or 4-sided dice?

Edit: I gave my idea for counting the distinct sequences for an equality to occur after exactly $n$ steps in this post: Number of ways a dice can roll every side equally many times for the first time after x rolls

Answer 1

With no further assumptions, it's zero rolls.

For every side to be rolled equally many times, with $k$ sides, you must roll a multiple of $k$ times, say $jk$ times. The condition is surely met with $j=0$. If the condition is ever to be met for larger $j$, at least two assumptions that you have not stated are needed:

• You must roll the die at least once.
• The probability of rolling a given side is neither zero nor one.

Without these assumption, we stand on $j=0$, the number of rolls is zero.