A radio show conducts a game according to the following rules. A number between

Question

A radio show conducts a game according to the following rules. A number between 1 and 100 is selected at random, and the disc jockey announces what number is chosen. Call it n. The announcer then randomly selects a number between 1 and n and asks that the first caller guess randomly selects a number between 1 and asks that the first caller guess the value of the second number chosen. Assuming that the caller's guess is equally likely to be any number between 1 and n, determine the expected value of the number guessed by the caller.

Explanation / Answer

First number chosen between 1 and 100 and announced as n

secnd number chosen between 1 and n

then the expected value of the number (i.e. second number) guessed by the caller is

E(x) =(1/n)*(1+2+3+.....+n)

=(1/n)*[n(n+1)/2]

=(n+1)/2

therefore, expected value of the number guessed by the caller is E(x) = (n+1)/2

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