Derive expected meetings given nonempty room

We have $N$ rooms, numbered $1$ through $N$, and $K$ meetings. Every meeting independently selects one of the rooms with equal probability. Let $K_1$ denote the count of meetings assigned to Room 1. Condition on the event $\{K_1 > 0\}$. (a) Find a closed-form expression for $E[K_1 \mid K_1 > 0]$ in terms of $N$ and $K$. (b) Find $Var(K_1 \mid K_1 > 0)$. (c) Briefly describe your method, for example using Bayes' rule or conditioning identities.

Overview: This question tests a candidate's grasp of conditional probability, zero-truncated binomial distributions, and computing conditional expectation and variance in finite-sample random assignment contexts.

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