Delhi School of Economics Course 003: Basic Econometrics
July 2015
Rohini Somanathan
Problem Set 1
1. Counting Methods and Basic Probability
(a) Suppose that one card is to be selected from a deck of 20 cards that contains 10 red cards and 10 blue cards, with cards of each colour numbered from 1 to 10. LetA be the event that a card with an even number is selected; let B be the event that a blue card is selected; and let C be the event that a card with a number less than 5 is selected. Describe the sample space S and each of the following events both in words and as subsets ofS
i. ABC ii. BCc iii. A∪B ∪C iv. A(B ∪C)
v. AcBcCc
(b) If k people are seated in a random manner in a row containing n seats n > k, what is the probability that they will occupy k adjacent seats? And if n people are seated in a random manner in a row containing 2n seats, what is the probability that no two people will occupy adjacent seats?
(c) Suppose that a deck of 52 cards contains 13 cards of each of 4 colors; red, yellow, blue and green. If the cards are distributed in a random manner such that each player receives 13 cards, what is the probability that each player will receive 13 cards of the same color?
(d) Suppose that there are 10 cards, 5 red and 5 green, and 10 envelopes, also 5 red and 5 green. The cards are placed at random in the envelopes. What is the probability that at least x envelopes will contain a card with a matching color (x= 0,1, . . . ,10)
(e) A coin is to be tossed as many times as necessary to turn up one head. Thus the elements of the sample space areH, T H, T T H, T T T H...etc. Write down a probability set function that assigns probabilities to these elements and show thatP(S) = 1, where S refers to the sample space.
(f) Three distinct integers are chosen at random from the first 20 positive integers. Compute the probability that their sum is even
(g) Players A and B play a sequence of independent games. Player A throws a die first and wins on a six. If he fails, B throws and wins on a five or a six. If he fails, A throws again and wins on a four,five or six, etc. Find the probability that Player A wins.
(h) Two fair dice are thrown. A is the event that the number on the first die is even and B is the even that the sum of the dice is odd. Are A and B independent?
(i) Prove that for every two eventsA andB, the probability that exactly one of the two events will occur is given by the expression P(A) +P(B)−2P(AB)
(j) Suppose that three runners from team A and three runners from team B participate in a race.
If all six runners have equal ability and there are no ties, what is the probability that the three runners from teamA will finish first, second and third?
2. Suppose thatA,B, andC are three events such that AandB are disjoint,Aand C are independent and B and C are independent. Suppose also that 4P(A) = 2P(B) =P(C)>0 andP(A∪B∪C) = 5P(A). Determine the value of P(A).
3. The Elpida Memory company produces memory chips on two different assembly lines, line 1 produces 80% of the company’s output and line 2 produces 20%. The probability that a chip is defective is .05 for the first line and .01 for the second.
(a) If a chip is chosen at random from the day’s production, and is found to be defective, what is the probability that it is made on line 2?
(b) The company invests in a testing device which can test for defective chips. P(A|B) = P(A∗|B∗) = .98, where A is the event that the testing device indicates that the chip is faulty, and B is the event that it is actually faulty. If the testing device is applied to the chips produced on line 1, what is the probability that the chip is faulty given that the testing device indicates that it is faulty? Suppose that the company would like P(B|A) to be .95. What is the value of r=P(A|B) that will ensure this accuracy if the test is applied to chips produced on assembly line 1?
4. A murder is committed. The perpetrator is either one or the other of two persons,X andY. After an initial investigation, both of these persons seem equally likely to have been the perpetrator. Further investigation reveals that the perpetrator has blood type A and we know that ten per cent of the population has this blood type. X is also know to have blood type A. What is the probability that X is the perpetrator?
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