Suppose you have two players $X$ and $Y$ fighting, both of which have $n\in \mathbb{N}, n\geq1$ life.

Each player has a probability $p_i$ of doing $i$ damage for all $i\in[0, n]$. Note that $p_0$ is the probability of doing no damage, and all $p_i$ should sum to 1. $X$ and $Y$ both share probabilities, so all $p_i$ for $X$ equals $p_i$ for $Y$.

Now play the game as follows (each step is one turn):

- Player $X$ attacks, and randomly deals $i$ damage to player $Y$ according to the distribution $p_i$. This means that $Y$ decreases their life by $i$.
- Player $Y$ attacks, and randomly deals $i$ damage to player $X$ according to the distribution $p_i$. This means that $X$ decreases their life by $i$.

The game ends when a player's life is $\leq 0$.

Now a person watching this game would like for it to end after about $t\in \mathbb{N}, t\geq 1$ turns. In what ways can they assign the probabilities $p_i$ such that the expected number of turns is $t$?

Edit: To be clear I am looking for all possible $p_i$ such that the expected number of turns is $t$.

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