Probability Problems and Solutions

Problem Sessions Richard J. La Spring 2018

Basic Probability - 1 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 7 Independents. How many different committees are possible?

Basic Probability - 2 A system consists of 5 components, each of which is either working or failed. Consider an experiment of observing the status of each component, and let the outcome of the experiment be given by the vector (x1, x2, x3, x4, x5), where the i-th entry is equal to 1 if component i is working and is equal to 0 if component i is failed. 1. How many outcomes in the sample space? 2. Suppose that the system will work if components 1 and 2 are both working, or if components 3 and 4 are both working, or if components 1, 3 and 5 are all working. Let W be the event that the system will work. Assume that each component i will work with probability , independently of each other. Compute the probability of W. 3. Let A be the event that components 4 and 5 are both failed. Compute the conditional probability

Basic Probability - 3 A pair of dice is rolled until a sum of either 5 or 7 appears. Find the probability that a 5 occurs first.

Basic Probability - 4 An urn contains 3 red and 7 black balls. Players A and B withdraw balls from the urn consecutively until a red ball is selected. Find the probability that A selects the red ball. (A draws the first ball, then B, and so on. There is no replacement of the balls drawn.)

Basic Probability - 5 A network manager (NM) has to decide whether there is a network anomaly or not based on observations. During normal operation, the NM will mistakenly determine that there is an anomaly with probability 0.2 (called false positive or type I error). Similarly, when there is an anomaly, the NM will miss it and determine that the network is operating normally with probability 0.05 (called missed detection or type II error). Suppose that the probability that there is a network anomaly is 0.1. Let A be the event that the NM declares a network anomaly. Compute the probability of A. Given that the NM declares a network anomaly, what is the probability that there is indeed a network anomaly?

Basic Probability - 6 Mary is planning to take the first three actuarial exams in the coming summer. She will take the first exam in June. If she passes that exam, then she will take the second exam in July, and if she also passes that one, then she will take the third exam in September. If she fails an exam, then she is not allowed to take any other exam. The probability that she passes the first exam is 0.9. If she passes the first exam, then the conditional probability that she passes the second exam is 0.8, and if she passes the first two exams, then the conditional probability that she passes the third exam is 0.7. What is the probability that she passes all three exams? Given that she did not pass all three exams, what is the conditional probability that she failed the second exam?

Discrete RVs 1 In college basketball, when a player is fouled while not in the act of shooting and the opposing team is in the penalty , the player is awarded a 1 and 1 . In the 1 and 1, the player is awarded one free throw and if that free throw goes in, the player is awarded a second free throw. Find the PMF of Y, the number of points scored in a 1 and 1 given that any free throw goes in with probability p, independent of any other free throw.

Discrete RVs 2 You are the manager of a ticket agency that sells concert tickets. You assume that people will call three times in an attempt to buy tickets and then give up. You want to make sure that you are able to serve at least 95% of the people who want tickets. Let p be the probability that a caller gets through to your ticket agency. What is the minimum value of p necessary to meet your goal. Each telephone ticket agent is available to receive a call with probability 0.2. If all agents are busy when someone calls, the caller hears a busy signal. What is the minimum number of agents that you have to hire to meet your goal of serving 95% of the customers who want tickets?

Discrete RVs 3 A radio station gives a pair of concert tickets to the sixth caller who knows the birthday of the performer. For each person who calls, the probability of knowing the performer s birthday is 0.75. All calls are independent. What is the PMF of L, the number of calls necessary to find the winner? What is the probability of finding the winner on the tenth call? What is the probability that the station will need nine or more calls to find a winner? Given that only one caller knew the correct birthday out of the first 5 callers, what is the conditional PMF of L?

Discrete RVs 4 The Sixers and the Celtics play a best out of five playoff series. The series ends as soon as one of the teams has won three games. Assume that either team is equally likely to win any game independently of any other game played. Find The PMF for the total number N of games players in the series The PMF for the number W of Celtic wins in the series Compute E[W]

Discrete RVs 5 At a discount brokerage, a stock purchase or sale worth less than $10,000 incurs a brokerage fee of 1% of the value of the transaction. A transaction worth more than $10,000 incurs a fee of $100 plus 0.5% of the amount exceeding $10,000. Note that for a fraction of a cent, the brokerage always charges the customer a full penny. You wish to buy 100 shares of a stock whose price D in dollars has PMF What is the PMF of C, the cost of buying the stock including the brokerage fee?

Discrete RVs 6 Suppose that a cellular phone costs $20 per month with 30 minutes of use included and that each additional minute of use costs $0.50. If the number of minutes you use in a month is a geometric random variable M with expected value of E[M] = 1/p = 30 minutes, what is the PMF of C, the cost of the phone for one month?

Multiple Discrete RVs 1 Suppose that the joint PMF of RVs N and K is given by Find the marginal PMFs of N and K Are N and K independent?

Multiple Discrete RVs 2 Suppose that an engineer inspects n circuits. Each circuit is acceptable with probability p, independently of each other. Let X be the number of acceptable circuits out of the n inspected circuits, and Y be the number of acceptable circuits found before observing the first reject (unacceptable). What is the joint PMF of X and Y? Are X and Y independent? Find the marginal PMFs of X and Y What is the conditional PMF of Y given {X = 1}?

Multiple Discrete RVs 3 Suppose that two RVs N and K have the joint PMF Find the marginal PMFs of N and K Compute the conditional expected value E[K | N = n] and conditional expectation E[K | N]

Continuous RVs 1 Suppose that the CDF of RV W is given by What is P(-2 < W < 2)? Find the PDF of W Compute E[W]

Continuous RVs 2 Let X be a RV with PDF Define Compute E[X] and Var(X) Find the CDF and PDF of Y Compute E[Y]