Solution
The Selective College Correlation
Show the problem again
Two fair six-sided dice are rolled independently, giving $(A, B)$. A pair is admitted only if $A + B \ge 10$. Among admitted pairs (each of the 36 outcomes equally likely beforehand), what is the correlation between $A$ and $B$? Give a reduced fraction.
Worked solution
The admitted outcomes are (4,6), (5,5), (6,4), (5,6), (6,5), (6,6): six outcomes. Over them $E[A] = E[B] = 32/6 = 16/3$, $E[A^2] = 174/6 = 29$, so $\operatorname{Var}A = 29 - 256/9 = 5/9$. Also $E[AB] = (24+25+24+30+30+36)/6 = 169/6$, so $\operatorname{Cov} = 169/6 - 256/9 = -5/18$. The correlation is $(-5/18)/(5/9) = -1/2$. Although $A$ and $B$ are independent in the full population, selecting on their sum induces a negative association (Berkson's paradox).
Source: Berkson's paradox (Berkson 1946); setup original. Statement written for AxiomIQ.