Question

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An urn model is known in the field of probability and statistics as a useful representation of a

probabilistic problem using coloured balls in an urn. There are many different variations of an

urn model. For this problem, consider the following case:

• There is an urn with 1 red ball and 1 blue ball.

• Every time a ball is drawn (at random) from the urn, it is placed back in the urn along with

2 more balls of the same colour as the ball that was drawn, and 1 more ball of the other

colour.

Denote the random variable Xi to be the number of red balls after the i-th drawn ball (for

i = 1, 2 . . .). Note that Xi is the random variable for the number of red balls in the urn including

the three new balls added after the i-th draw.

(a) Find the probability mass function (pmf) of X2.

(b) What is the probability that the first ball drawn was red, given that there are at least 5 red

balls after the third ball is drawn.

(c) Compute E(X3) and Var(X3)

Answer 1