1. [10 points] In a shop, 20% of the customers spend at least 1000 dollars. Among those spending at least 1000 dollars, 30% pay by credit card, while among those spending less than 1000 dollars, only 10% pay by credit card. Suppose a randomly chosen customer is found to have paid by credit card. 'What is the probability that he or she spent at least 1000 dollars? 2. [10 points} A clam of introductory probability consists of 10 sophomores, 60 juniors and 30 seniors. The nal grades show that 5 sophomores, 9 juniors and 1 senior reoeived an A grade. Assume B, C and D are the events that the student is a sophomore, junior and senior student, respectively. Let A be the event that the student got an A grade. (a) If a randomly chosen student is found to have earned an A grade, nd the probability that he or she is ajlulior, that is P(CEA). (b) Find P(D) and P(D[A). {o} Are \"getting an A grade" and \"being a junior" independent events? How about the events of \"getting an A grade\" and \"being a senior\"? 3. [10 points] The probability mention L: for a random variable X is given by the table below. (9.} Find u. (h) Find the expected value and vmianee of the distribution for the random variable X : E(X ) and Var(X ). {1:} Assume now 1\" = X 2, then you may regard 1" as another random variable. Find the expected value and variance for the distribution of this random variable Y: Eli?) and Var']. (d) No need to answer this part. Just a remark for your interest: In part (c), you are nding E(X 2] and Var(X 2). It would be interesting to compare these values with ELY} and Var(X]. 4. [10 points} A sale-5 manager receives 6 telephone calls cn average between 9 : 303m and 10 : 30am on a weekday. Find the probability that (a) the manager will receive 2 or mere calls between 9 : 30am and 10 : 30am cn a certain weekday. (b) the manager will receive exactly 2 calls between 9 : 30am and 9 : 40am. (e) during a normal 5-day working week. there will be exactlyr 3 days on which the manager will receive exactly three calls between 9 : 30am and 9 : 40am. 5. [10 points] Suppose that X ~ N [10. 16} with mean 10 and variance 16. Find (a) P(9.6 S X 11):: 0.5517. (c) a. an that For