1. [Conditional Probability and Independence] (a) Consider events A, B, 0' related to a. particular random experiment. True or False: If P{A|C') :- P{B|C] and P[A|Cc] \"a- P{B|C\"] then PTA) } FIB). (b) Consider tossing a fair coin independently twice. Dene the events: a A = {the outcome of the rst toss is T}, II B = {the outcome of the second toss is T}, a C = {the outcome of both tosses is the same}. Are A, B. C pairwise independent? Are they independent? 2. [Binomial Distribution] Let X be a binomial random variable with parameters 51 and 53. Use LOTUS to show that 3. [Poisson Distribution] Let X be a Poisson random variable with parameter A. (a) Show that P{X = 23m = 0,13,...) = %[1 + 5\"]. (b) True or False: E[X3] = AE[{X | If]. 4. [Probability in Games] Suppose that a roulette wheel consists of the numbers {0. 1,2, . .. .36} and the additional 'D'. Suppose that you always bet that the outcome will be in the set {1,'2, . .. .15}. 1What is the probability that (a) you lose your rst 3 bets. (b) your rst win occurs on the 5th bet. 5. [Theoretical Extensions] Consider performing independent trials, each with probability of success 13 E (D. 1). until accumulating in successes. Let X be the corresponding number of trials until then. It can be shown that 51] 1395:\"): (k_l)p'=(1_p]"", n=k,lc+1.... In this setting, what is the probability of it successes occurring before in failures