Question:We have a bag that contains n red balls and n blue balls. At each of 2n rounds we remove one of the balls from the bag randomly, and place it in one of available n bins. At each round, each one of the balls that remain in the bag is equally likely to be picked, as is each of the bins, independent of the results of previous rounds. Let N(k) be the number of balls in the k-th bin after 2n rounds, i.e., after all balls have been placed in the bins.

a) Find the probability that N₁ = 0, i.e., that the first bin is empty after all balls have been removed and placed into bins?

b) What is the PMF of p_N1(k) of N₁?

c) What is the expected number of empty bins?

d) What is the probability that the ball picked in the third round is red?

e) Let R_i denote the event that i-th ball picked is red. Are the events R₁ and R₂ independent?