Take the 2-minute tour ×  Suppose I pick $k$ integers without
replacement from $\{1, \ldots, n\}$. Let $I$ be the value of the
highest integer. A calculation with binomials reveals $$E[I] =
\frac{k}{k+1}(n+1)$$ This is a very simple formula - does it
have a simple calculation-free proof?  share|improve this
question Do you know some instance of calculation-free proof for
the expected value of some non-constant random variable? I would
like to see it. –  Matemáticos Chibchas Feb 7 '13 at 8:22 In
many cases, you can use symmetry or some clever conditioning to
avoid heavy calculations. –  pierre Feb 7 '13 at 8:33 You just
said it: "heavy" calculations. This is OK, but no calculations
at all? Besides, I agree that conditioning is clever, but it is
based on the nontrivial subject of conditional expectation. – 
Matemáticos Chibchas Feb 7 '13 at 9:04 @MatemáticosChibchas -
the answer by Henry below is what I would call calculation-free.
–  pierre Feb 8 '13 at 23:42 add comment  1 Answer  up vote 4
down vote accepted