# Conditional Probability

Do Problem 2 ONLY please.

Problem 1

An urn contains n+m balls, of which n are red and m are black. They are withdrawn from the urn, one at a time and without replacement. Let X be the number of red balls removed before the first black ball is chosen. Find E[X]. To obtain this quantity, number the red balls from 1 to n. Now define the random variables Xi , i=1,....,n by

Xi = 1 if red ball I is taken before any black ball is chosen

0 otherwise

a) Express X in terms of the Xi

b) Find E[X]

Problem 2

From the above problem, let Y = number of red balls chosen after the first but the second black ball has been chosen

a) express Y as the sum of n random variables, each of which is equal to either 0 or 1

b) find E[y]

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#### Solution Preview

a) Let Yi be a randon variable which is 1 if the ith ball drawn after the first black ball ...