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Let $m\ge1$, $n\ge1$, and $k\ge1$ be integers with $k\le m + n$. Consider a set $P$ consisting of $m$ men and $n$ women. We choose a uniformly random $k$-element subset $Q$ of $P$. Consider the random variables

  • $X$ = the number of men in the chosen subset $Q$,
  • $Y$ = the number of women in the chosen subset $Q$,
  • $Z = X − Y$.

Determine the expected value ${\rm E}(X)$.

Prove that ${\rm E}(Z) = k\dfrac{m - n}{m + n}$.

Nicolas FRANCOIS
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ajohnsy96
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  • What have you tried so far? Where are you stuck? What kind of things have you learned recently that might be useful tools in tackling this problem? – ConMan Apr 03 '19 at 22:42
  • As a small hint - note that Y = k - X and hence Z = 2X - k, so once you figure out one expectation you can use linearity of expectations for the other two. – ConMan Apr 03 '19 at 22:43
  • Welcome to Math.SE. Please use MathJax when writing equations to make them easier to read. – Ertxiem - reinstate Monica Apr 03 '19 at 22:44

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