please provide explanations for each below

1. [26 Points] Independence and Expectation. An urn contains 10 balls: 5 white balls numbered 1 through 5, a further 3 black balls numbered 1 through 3, and 2 red balls numbered 1 through 2. We simultaneously and randomly draw 2 balls out of the 10. (a) Find the probability of event A: the two balls are white. (b) Find the probability of event B : the two balls are odd. ((3) Are events A and B independent? Prove your answer. (d) Let X be the random variable whose value is the number of white balls in the drawing. i. Write the probability distribution of X in form of a table: ii. Find the expected value E[X]. (e) Suppose that you are to win $1 for each white ball that is drawn, $2 for each black ball, and $3 for each red ball. What is your expected gain from this game? Use 10 indicator variables W1,W2, . . . ,W5,Bl, BQ,B3,R1,R2 , one per hall. For example variable W1 is 1 if white ball 1 is in the drawing and 0 otherwise. Find the expectations of these variables and then use linearity of expectation to compute your expected gain. (Explain how and where linearity of expectation was used in your calculation.)