A coin is tossed repeatedly (infinitely many times). Let A denote the event that the first head occurs after the result of the 5th toss is recorded. Let B denote the event that the first head has not occurred in the first 2 tosses. That is, A = "first head after the 5th toss"; and B = "no head in first 2 tosses." Find the following probabilities. Give each using 3 decimal places. Pr(A n B) = la. Pr(A n B): Pr(B) = 1b. Pr(B): Pr(A | B) = 1c. Pr(A | B): Pr( B | A) = 1d. Pr(B | A):2. Find a formula for the probability that among a set of )1 people, at least two have their birth- days in the same week of the year (assuming 52 weeks per year, and assuming all weeks are equally likely for birthdays). Assume n _<_1 n> 52 the probability is equal to 1.0). Let A ="at least two matching birthday weeks\3. Independent trials that result in a success with probability p are successively performed until a total of r successes is obtained. Let X = number of trials that are needed (total number, including the r successes). In the following questions please mark the correct answer. Use q = 1 p. 351. Find the probability function pn) = Pr(X = n). (Him: the r-th success must be the last trial, but the other r 1 successes can be any trial). (a) 19'4"" (b) (3);)?" (c) (":l)P'q"' (d) (tnp'q'H (e) (::]')p"lq"" (f) none of these 31). Find E(X). Hint: You might nd it useful to use a geometric series with 0