Please include all steps. Thanks An urn model is known in the field of probability and...

Please include all steps. Thanks

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.

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