100A Final Exam Review ● Permutations, Combinations, Binomial & Multinomial Theorem (12) ● Probability laws (3  4) ● Expected value and variance (5  6) ● Discrete distributions (7  12) ● Continuous Random Variables (1214) ● MGF (11, 13d) ● Conditional Expectation and Variance, Covariance, Joint Distributions (15  16) ● Transformations of Random Variable (17) 1) Use the binomial theorem to show that a)nCk (1)k =0∑nk=0 b)nCk (a1)k =an∑nk=0 2) a) (1.18) A committee of 7, consisting of 2 Republicans, 2 Democrats, and 3 Independents, is to be chosen from a group of 5 Republicans, 6 Democrats, and 4 Independents. How many committees are possible? b) (1.29) Ten weightlifters are competing in a team weightlifting contest. Of the lifters, 3 are from the United States, 4 are from Russia, 2 are from China, and 1 is from Canada. If the scoring takes account of the countries that the lifters represent, but not their individual identities, i) how many different outcomes are possible (in terms of final rankings, assuming no ties) ? ii) How many different outcomes correspond to results in which the United States has 1 competitor in the top three and 2 in the bottom three? c) (2.44) Five people, designated as A, B, C, D, E, are arranged in linear order. Assuming that each possible order is equally likely, what is the probability that i) there is exactly one person between A and B? ii) there are exactly two people between A and B? iii) there are three people between A and B? d) (2.35) Seven balls are randomly withdrawn from an urn that contains 12 red, 16 blue, and 18 green balls. Find the probability that i) 3 red, 2 blue, and 2 green balls are withdrawn; ii) at least 2 red balls are withdrawn; iii) all withdrawn balls are the same color; iv) either exactly 3 red balls or exactly 3 blue balls are withdrawn. 3) a) Suppose that A, B, and C are three events such that A, B are disjoint, A, C are independent and B, C are independent. Suppose also that 4P(A) = 2P(B) = P(C), and P(A B C) = 5P(A). Determine the value of P(A).⋃ ⋃ b) Two events, A and B, are independent. Show that A and Bc are also independent. 4) a) (2.9) A retail establishment accepts either the American Express or the VISA credit card. A total of 24 percent of its customers carry an American Express card, 61 percent carry a VISA card, and 11 percent carry both cards. What percentage of its customers carry a credit card that the establishment will accept? b) Ninety percent of all babies survive delivery. However, 15 percent of all births involve Csections, and when a C section is performed the baby survives 92 percent of the times. If a randomly chosen pregnant woman does not have a C section, what is the probability that her baby survives? 5) Two marbles are taken, one at a time, without replacement, from an urn which has 6 red and 10 blue marbles. We win $2 for each red marble chosen and lose $1 for each blue marble chosen. Let X be the winnings/losings. a) What is the chance both marbles are red? b) What is the distribution of the number of red marbles selected? c) What is the PMF of X? d) E(X)? e) Var(X)? 6) Let M = the maximum value when two dice are rolled. Find: a) E(2M  M^2) b) Var(3M +5) 7) Consider the casino roulette game. If a player bets $1 on a single game and we let X be the casino’s profit we get the following probability distribution. X P(x) 35 1/38 1 37/38 In 500 such games, what is the exact probability that the casino will make more than $200? You don’t need to compute the probability, just write the exact expression for it. 8) The probability of receiving a onepair poker hand (e.g., 2,3,King,10,10) is 42%. Find the probability that the 6th onepair poker hand for a player will be observed on his 10th game. 9) Let X follow the binomial distribution with parameters n and p>0. Is it true that the mean is always larger than the standard deviation? If not, please find the condition under which this is not true. 10) A (weighted) coin has a probability of p = 0.7 of coming up heads (and so a probability of 1 − p = 0.3 of coming up tails). This coin is flipped until a head comes up. a) What is the expected number of flips? b) Suppose the number of flips is limited to 4. What is the expected number of flips? What is the probability of heads coming up? 11) a) Show that the distribution of the sum of two independent, poissondistributed random variables is poissondistributed b) Suppose the number of people who live to 100 years of age in Westville per year has a Poisson distribution with parameter λ1 = 2.5; whereas, independent of this, in Michigan City, it has a Poisson distribution with parameter λ2 = 3. What is the chance that the sum of the number of people who live to 100 years of age in Westville and Michigan City is 4? 12) Suppose that the number of cars arriving at a stop sign in any given minute is poissondistributed with mean 4, and behavior from minute to minute is independent. (For help see week 7 handout, “Poisson Gamma and Exponential Distribution”) a) You walk up to the corner. What is the probability of no cars arriving in the next minute? b) What is the probability of no cars arriving in the next 30 seconds? c) What is the probability that you will wait at least three minutes before seeing a car? d) What is the probability that you will have to wait less than 5 minutes before 3 cars arrive? e) What is the probability that you will wait exactly 5 minutes before 3 cars arrive? (or, more precisely, the value of the pdf for wait time of 5 minutes before 3 cars) 13) Let X be an exponential random variable with lambda = 1/2. Find the a) mean of X b) median of X c) variance of X d) E(X3) 14) A parachutist lands at a random point on a line from point A to point B. a) Find the probability that his distance to A is more than 3 times his distance to B b) Now suppose that his target is halfway between point A and point B and his distance from the target (still on the line AB) is normally distributed with variance (AB)2/16. What is the approximate probability that he lands between A and B? 15) (6.16 self test) You and three other people are to place bids for an object, with the high bid winning. If you win, you plan to sell the object immediately for 10 thousand