There is an urn containing 9 balls, which can be either green or red. The number of red balls in the urn is not known. One ball is drawn at random from the urn, and its color is observed.

(a) What is the Bayesian universe of the experiment.

(b) Let X be the number of red balls in the urn. Assume that all possible values of X from 0 to 9 are equally likely. Let Y1 =1 if the first ball drawn is red, and Y1=0 otherwise. Fill in the joint probability table for X and Y1 given below:

X prior Y1 = 0 Y1 = 1

(c) Find the marginal distribution of Y1 and put it in the table.

(d) Suppose a red ball was drawn. What is the reduced Bayesian universe?

(e) Calculate the posterior probability distribution of X.

(f) Find the posterior distribution of X by filling in the simplified table:
X prior likelihood prior × likelihood posterior marginal P(Y1 = 1)