 Marginal and pair probabilities in sampling without replacement

Take a uniformly random permutation of $N>1$ distinguishable objects, of which $K$ are marked. Let $B_i$ mean that position $i$ is marked. Every object is equally likely at a specified position, so $\mathbb{P}(B_i)=K/N$. For distinct positions $i,j$, their ordered pair is uniform among $N(N-1)$ distinct-object pairs, of which $K(K-1)$ are marked pairs. This proves the second formula. For $K>0$, the conditional probability is $\mathbb{P}(B_i\mid B_j)=(K-1)/(N-1)$.

For the count $M_m$ of marked objects among the first $m$ positions, indicator random variables and linearity of expectation give $\mathbb{E}[M_m]=mK/N$. These draws are exchangeable but generally dependent; equal marginal probabilities do not imply independence.