5-56. (Bayes’ Rule for discrete random variables) Using (5.4) and (5.13), express

(5.9) as g(X|Y) = g(X)h(Y|X)

i g(X)h(Y|X)

.

Similarly, show that (5.11) can be written, via (5.3) and (5.13), as h(Y|X) = h(Y)g(X|Y)

i h(Y)g(X|Y)

.

Let vessel A1 contain one red (R) marble and two white (W) marbles and let vessel A2 contain three red marbles and one white marble.

Suppose a marble is randomly drawn from vessel A1 and placed into vessel A2. Call its color the value of the random variable Y. Then a marble is randomly selected from vesselA2. Call its color the value of random variable X. If X = W, what is the probability that Y = R? (Hint: Find h(R|W). )