{title2=$\pi_{ij}=\mathbb P(i,j\in S)$}

A <joint inclusion probability> is the probability that two specified population units both belong to a random sample $S$:
$$
\pi_{ij}=\mathbb P(i\in S,\ j\in S).
$$
For the same unit twice, $\pi_{ii}=\pi_i$, where $\pi_i$ is its <inclusion probability>. For distinct units, $\pi_{ij}$ is generally not $\pi_i\pi_j$: selection can make their <inclusion indicators> dependent. These probabilities describe the sampling procedure, rather than a relationship between the measured values of the two units.<Unequal Probability Sampling in Penn State STAT 506>[Lesson 3, inclusion probabilities and finite-population estimation]{ref}

For <simple random sampling> of $n$ distinct labels from $N>1$ labels, counting subsets containing both $i$ and $j$ gives
$$
\pi_{ij}=\frac{\binom{N-2}{n-2}}{\binom Nn}
=\frac{n(n-1)}{N(N-1)},\qquad i\ne j,
$$
with value zero if $n<2$. Since $\pi_i=n/N$,
$$
\operatorname{Cov}(I_i,I_j)=\pi_{ij}-\pi_i\pi_j
=-\frac{n(N-n)}{N^2(N-1)}.
$$
For $N=4,n=2$, each unit has <inclusion probability> $1/2$, but a particular pair has <joint inclusion probability> $1/6$. Conditional on including one member, the other is included with probability $(1/6)/(1/2)=1/3$, not $1/2$.

Equal individual <inclusion probabilities> do not determine these pair probabilities. A different procedure that chooses only $\{1,2\}$ or $\{3,4\}$, each with probability $1/2$, still includes each unit with probability $1/2$. Yet $\pi_{12}=1/2$ and $\pi_{13}=0$. Thus two procedures can agree on individual representation while producing different dependence and different precision for a population summary. The distinction motivates keeping pair information when calculating the <variance> of weighted survey estimators; the classical general treatment is due to <D. G. Horvitz> and <D. J. Thompson>.<A Generalization of Sampling Without Replacement From a Finite Universe by D. G. Horvitz and D. J. Thompson>[Publisher abstract and original publication, pp. 663–685]{ref}


 Joint inclusion probability (source code)