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An urn contains a white and b black balls After

An urn contains a white and b black balls. After a ball is drawn, it is returned to the urn if it is white; but if it is black, it is replaced by a white ball from another urn. Let Mn denote the expected number of white balls in the urn after the foregoing operation has been repeated n times.

(a) Derive the recursive equation

Mn+1 = (1 – 1/a + b) Mn + 1

(b) Use part (a) to prove that

Mn = a + b – b (1 – 1/a + b)n

(c) What is the probability that the (n + 1)st ball drawn is white?

(a) Derive the recursive equation

Mn+1 = (1 – 1/a + b) Mn + 1

(b) Use part (a) to prove that

Mn = a + b – b (1 – 1/a + b)n

(c) What is the probability that the (n + 1)st ball drawn is white?

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