Conditional Expectation (N coin flips)

Question

Looking at $N$ coin flips with a probability of $p \in (0,1)$ with the probability measure

$P(X_1 = x_1, \dots, X_N = x_N) = p^k (1-p)^{N-k}$ with $k=x_1+\dots +x_N$
where $X_i(\omega) = x_i$ for $\omega = (x_1, \dots, x_N) \in \Omega = \{0,1\}^N$

I want to figure out the conditional expectation of

$E(X_i|S_n = K) \quad (i=1, \dots N)$
where $S_N:= \sum_{i=1}^{N}X_i$ and $K$ being the number of outcomes with the probability $p$.

Answer 1

$E(X_i|S_N=K)$ is the same for all $i$ and the sum of these expectations is $$ \sum_{i=1}^N E(X_i|S_N=K)=E\left(\sum_{i=1}^NX_i\,\Big| S_N=K\right)=E(S_N|S_N=K)=K.$$ We get therefore $E(X_i|S_N=K)=\frac KN$.