We arrange $9$ balls numbered $1,\dots,9$ in a row randomly (a permutation). Let $X_i$ an indicator to the ball in the $i$ position is less than the ball in the $i+1$.
Prove $\mathbb{E}[X_2|X_1]=\frac{1}{3}(2-X_1)$
I would appreciate any direction.
2026-03-27 10:08:14.1774606094
$\mathbb{E}[X_1|X_2]$ of permutation on $9$ elements
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Ignore the balls in positions $4$ to $9$. There are $3!=6$ different rankings of the balls in positions $1$ to $3$, and they're equiprobable by symmetry. Then it's just a matter of counting cases.