Expected value of finding the second ball drawn question

Question

Please show me the steps to find the answer to part b

Three urns are numbered 1 through 3; urn k contains k balls numbered 1 through k. We select an urn at random, draw a ball from it, note the number of the ball, replace the ball, and then draw again from the same urn. If it is known that the first ball drawn has number 1, find (a) the probability mass function of the number of the selected urn; (b) the expected value of the number of the second ball drawn.

I found the answer to part (a) but i am struglling with part (b) finding the expected value of the number of the second ball drawn. the answer to part (b) is 29/22 but i dont know how you get that answer.

Answer to part (a) : (k=1, 6/11) (k= 2, 3/11) ( k =3 , 2/11)

So pleae explain how to get the answer

Answer 1

You draw with replacement, so distribution of the first and second balls are identical. What is the probability you draw ball labeled 2?

$$1/3*0 + 1/3*1/2 + 1/3*1/3 = 5/18$$

Can you find probability to draw ball labeleld 1 and labeled 3? Now, what is expected value of the ball to be drawn?

Answer 2

Let random variable $X$ be the number on the second ball drawn.

If the urn was Urn $1$, then $E(X)=1$. More formally, let $Y$ be the number of the urn. Then the conditional expectation $E(X|Y=1)$ is $1$.

If it was Urn $2$, then $E(X)=\frac{3}{2}$.

If it was Urn $3$, then $E(X)=\frac{6}{3}$.

It follows that $E(X)=(6/11)(1)+(3/11)(3/2)+(2/11)(6/3)$.