Problem 5. [Hard] For this problem, you might need to use the following so- called inclusion-exclusion principle without proof. Let Aj, Ay, ... , A, be n events. Then P(A, U . . . UA.) = EP(A.) - _ P(AinA,) + P(AnA, nAR) -...+ (-1)" 'P(A, n.. . nA.). A little girl is painting on a blank paper. Suppose that there is a total number of N available colors. At each time she selects one color randomly and paints on the paper. It is possible that she picks a color that she has already used before. Different selections are assumed to be independent. (1) Suppose that the littile girl makes n selections. (1-i) If red and blue are among the available colors, let R (respectively, B) be the event that her painting contains color red (respectively, blue). What is P(R) and P(RUB)? (1-ii) Suppose that she is about to make the (n + 1)-th selection. What is the probability that she will obtain a new color in this selection? [Hint: discuss according to the specific color in her (n + 1)-th selection.] (1-iii) Suppose that n = N. For 1 n} and use the inclusion-exclusion principle to compute this probability.) (2-ii) For 0 K) given in Tutorial 6, Problem 4 to simplify the expression of the mean. Secondly, relate E[T] with either some standard Taylor series to guess the growth rate or relate it with Poisson random variables and the central limit theorem. The central limit theorem (which we will learn soon) says, if X1, X2, . .. are independent and identically distributed with finite variance, then lim P Sn - ES cr = $(2), for all I ER, Var[S..] where S.. = X1 + ...+X, and (z) is the Cdf of the standard normal distribution.]