In a box there are $k$ white and $l$ black balls, and next to the box there are $m$ white and $m$ black balls. Out of the box one ball is randomly taken out and then returned to the box also with $m$ more balls of the same color.
a) Find expected value of the number of white balls that are now in the box
b) If now we are picking a ball out of the box, what is the probability that it will be white?
b) Hypotheses:
$H_{1}:$ I have taken a white ball out of the box
$H_{2}:$ I have taken a black ball out of the box
Let A be the probability we are looking for:
$P(A)=P(A\setminus H_{1})P(H_{1})+P(A\setminus H_{2})P(H_{2})=\frac{k+m}{k+m+l}\frac{k}{k+l}+\frac{k}{k+m+l}\frac{l}{k+l}$
a) I am not really sure how to solve this, I was thinking of a random variable $X$ that will represent number of white balls in a box, and it can only take values $k$ or $k+m$?
You are on the right track and can apply:
$$\mathbb EX=\mathbb E[X\mid H_1]P(H_1)+\mathbb E[X\mid H_2]P(H_2)$$
Just as you did with finding $P(A)$.