STAT 67, Winter 2017 Homework 3 2-07-2016 Show work for all parts of problems. 1. 5 points each. State yes/no in each part, and explain in a sentence or two. a. You have a bowl filled with 100 balls, which are colored red or blue. You have 50 red and 50 blue balls. You randomly select a ball and throw it out (don't put it back in the bowl). Then you go to randomly select another ball. Is the probability that the second pick is blue independent of what the first ball color was? b. You have a bowl filled with 100 balls, which are colored red or blue. You have 50 red and 50 blue balls. You randomly select a ball, record the color, and put the ball back into the bowl. Then you go to randomly select another ball. Is the probability that the second pick is blue independent of what the first ball color was? c. A basketball player is going to shoot the ball 25 times. They are interested in the number of shots they make (each shot is either made or missed). If each shot attempt is independent with a 95% chance (0.95 probability) of being made, then is the total number of shots made in the 25 attempts following a binomial distribution? d. A basketball player is going to shoot the ball 25 times. They are interested in the number of shots they make (each shot is either made or missed). If shot attempts are independent, but the player begins to shoot better the more they attempt shots (the probability they make a shot increases the more attempts they take, p is increasing), then is the total number of shots made in the 25 attempts following a binomial distribution? 2. 5 points each. A shuttle operator has sold 20 tickets to ride the shuttle. All passengers (ticket holder) are independent of each other, and the probability that a passenger is part of the frequent rider club is 0.65 (65% chance they are part of the group and 35% chance they are not). Say X is the number of passengers out of the 20 that are part of the frequent rider club. a. Was is the distribution of X? Write the pmf (f (x)), and name its parameters. What key i assumption of the passengers is needed to determine this distribution? b. What is the expected number of passengers that are part of the frequent rider club. Interpret this value for the shuttle operator (in a sentence or two). c. Say each passenger is charged $3.00 for their tickets. What is the expected revenue? What is the variance of the revenue? d. Now say each passenger in the frequent rider club are charged $2.00 for their tickets and the regular passengers (not in the frequent rider club) are charged $4.50 for their tickets. What is the expected revenue? 3. 5 points each. A dart is thrown at a target. The probability the dart will hit the target (yes or no) on a single attempt is 0.20. Each throw is independent of the other throws. Let X be the number of attempts before the target is hit. a. What distribution does X follow? State the name and parameters. b. Compute the expected number of attempts needed before hitting the target. c. What is the variance and standard deviation of the number of attempts needed before hitting the target? d. What is the probability that it will take less than 5 throws before hitting the target? Compute this value using R and state what function you used. 4. 5 points each. In proof testing of circuit boards, the probability that any particular diode will fail is 0.01. Suppose a circuit board contains 200 diodes, all which are independent of one another. A board works properly only if all its diodes work. a. How many diodes would you expect to fail, and what is the standard deviation of the number that will fail? b. If five boards are shipped to a particular customer, how likely is it that at least four of them will work properly? (A board works properly only if all its diodes work and assume the boards are independent of one another.) 5. 5 points each. Cars arrive at an inspection station according to a Poisson process at a rate of 10 cars per hour. With probability of 0.5, a car will pass inspection. a. What is the expected number of cars to arrive at the inspection station in 1 hour? What is the expected number of cars to arrive in a 3 hour period? b. What is the probability that at most (less than or equal to) 15 cars will arrive in the 1 hour time ii period? What is the probability that at most 45 cars will arrive in the 3 hour time period? c. In a 3 hour time period, what is the probability that more than 4 cars will arrive? d. Say the cars are independent of one another. What is the probability that 10 cars will arrive in a 1 hour time period AND all 10 will pass inspection. Hint: P(A and B)=P(A)*P(B|A) 6. 5 points each. How academic programs work are that students send an application. The student is then accepted or rejected by the department. If accepted, the student can choose to accept the offer or reject. Suppose an academic department has accepted 1000 students, but only has space for 800 of them. Let the accepted students be independent of one another, and the chance that a given student will accept the offer is 0.80. a. What is the probability that the department will have space for all the students that accept their offer? b. Now say we go down the list of students that the department accepted (the list of 1000). Starting with the first name, and going in order (1 to 1000), let X be the number of names (students) you go through before you reach the first one that accepts the offer. What distribution does X follow, and what is the parameter? 7. Not required, for your own practice. You are a contract vendor who gets hired to sell toys at a state fair. The fair organizers pay you 1 dollar a day and you keep any money you make from selling toys. You sell the toys for 2 dollars each. The number of toys you sell on a given day is Xi , which has expectation 10 and variance 1. You are hired for 5 days. Thus on the i-th (i=,2,3,4,5) day you make 1+2*Xi amount of dollars, where Xi is the number of toys you sell on that day. a. What is the expected amount of income (or revenue) you make on a single given day? b. What is the expected amount of income (or revenue) you make over the 5 days? c. What is the variance of the amount of the income you make over the 5 days