Q.312. Probability - Expectation.

Question 312.
Calculate the expected value of the greater of two numbers when two different numbers are picked at random from the numbers 1.......n.

Answer 312.
Number "n" can be selected with lower numbers in (n-1) ways.
Probability of selecting "n", P(n) = (n-1)/nC2
Probability of selecting (n-1) = (n-1)(n-2)/nC2
=> Expectation
= [n(n-1) + (n-1)(n-2) + (n-2)(n-3) +...+ 2*1] / nC2
= [Σ n(n-1) (n=2 to n)] / nC2
= [Σn^2 - Σn ... (n=2 to n)] / nC2
= [(1/6)n(n+1)(2n+1) - 1 - n(n+1)/2 +1]/nC2
= [(n+1)(2n+1) - 3(n+1)] / [3(n-1)]
= (2n^2 - 2) / [3(n-1)]
= 2(n+1)/3.

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