43. 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 They are withdrawn from the urn, one at a time and without replacement.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. We are interested in determining E[X]. To obtain this quantity, number the red balls from 1 to n. Now define the random variables Xi , i = 1, . . . , n, by

(a) Express X in terms of the Xi .

(b) Find E[X].

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x- if red ball i is taken before any black ball is chosen otherwise