Question
Two digits \(X\) and \(Y\) are drawn independently and uniformly from \(\{1,2,3\}\). An analyst keeps only selected pairs. Consider two separate screening rules:
Rule A: keep the pair when \(X\in\{1,2\}\) and \(Y\in\{2,3\}\).
Rule B: keep the pair only when it is one of \((1,1), (1,2), (2,2), (3,3)\).
Both rules retain four of the nine equally likely original pairs. After selection, are \(X\) and \(Y\) still independent under each rule? Demonstrate your conclusions using the conditional joint probabilities and marginals. Why is the shared 4/9 acceptance rate not enough to answer?
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Hint 1
Every accepted ordered pair has conditional probability 1/4, since all nine were initially equally likely and exactly four are retained.
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Solution
Before selection, all nine ordered pairs in \(\{1,2,3\}^2\) have joint probability \(1/9\). Under either rule, the selection event consists of exactly four pairs, so a selected pair has conditional probability \((1/9)/(4/9)=1/4\) at each accepted cell and zero elsewhere. Independence after selection requires the entire conditional table to factor as \(\Pr(X=x\mid S)\Pr(Y=y\mid S)\) for every \((x,y)\).
Rule A: The accepted cells are \((1,2),(1,3),(2,2),(2,3)\), exactly the rectangle \(\{1,2\}\times\{2,3\}\). Given A, \(\Pr(X=1)=\Pr(X=2)=1/2\) and \(\Pr(Y=2)=\Pr(Y=3)=1/2\). Every accepted cell has probability \(1/4=(1/2)(1/2)\), and every rejected cell has at least one zero marginal. Hence all cells factor and \(X\) and \(Y\) remain independent conditional on A.
Rule B: The accepted cells are \((1,1),(1,2),(2,2),(3,3)\). Given B, the conditional X marginals are \((1/2,1/4,1/4)\) and the conditional Y marginals are \((1/4,1/2,1/4)\) for values 1, 2 and 3. For example, \((X,Y)=(1,3)\) is never accepted, so \(\Pr(X=1,Y=3\mid B)=0\), yet
\[\Pr(X=1\mid B)\Pr(Y=3\mid B)=\tfrac12\cdot\tfrac14=\tfrac18>0.\]
Factorisation fails, so the variables become dependent after selecting on B. This is selection-induced dependence: a coupled joint inclusion rule can link originally independent entries. The probability of being accepted (4/9) is identical under A and B, but acceptance geometry—rectangular versus coupled support—is what determines the conditional independence verdict.
Assumptions
X and Y are independent and each uniform on the three values 1, 2 and 3.
Each rule defines a separate conditioning experiment; there is no additional random selection after the rule is applied.
Ordered pairs are distinguishable and conditioning is on the respective nonempty acceptance event.
Final Answer
Independent conditional on Rule A; dependent conditional on Rule B, despite both retaining 4/9 of the original pairs.