Research Methods in Economics
Research Methods in Economics ECO 320
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Date Created: 10/11/15
III PROBABILITY 31 INTRODUCTION It has been remarked that death and taxes are the only things that are certain in life While an obvious exaggeration the remark does serve to highlight the fact that in many activities in life one does not face certainty or perfect knowledge but rather uncertainty or partial knowledge Situations involving uncertainty range from the simple to the very complicated When one ips a coin one is uncertain about whether a head or a tail will be obtained When a person starts a new company there is uncertainty about whether it will survive nancially for at least ve years We shall see many other examples later What is important for now is to recognize that uncertainty is a pervasive phenomenon in our world Furthermore rational decision making be it about smoking sunbathing or about how much of a good a rm should produce next month requires that one be able to reason about uncertainty In particular one needs to be able to make quantitative assessments of the chances of events occurring Probability is that body of knowledge that deals with reasoning deductively about uncertainty As in any area of intellectual activity the theory of probability has a vocabulary that is used in its discussions Let us consider the following example involving uncertainty A class consisting of one hundred students is made up often freshmen twenty ve sophomores forty ve juniors and twenty seniors A person is selected at random and his class or year is recorded There are four possible outcomes The probability of drawing a freshman is 10100 the probability of getting a sophomore is 25 100 the probability of a junior is 45100 and the probability of obtaining a senior is 20100 Let us look at the foregoing example in detail We have a process by which observations are generated and recorded This is what statisticians call an EXPERIMENT Let us indicate this explicitly as follows EXPERIMENT 1 A class consists of one hundred students Ten are freshmen twenty ve are sophomores forty ve are juniors and twenty are seniors A person is selected at random and his class or year is recorded Note that the term experiment is used here to describe any process by which data or observations are generated and recorded Thus the term is used in a very broad sense What we have also in the above example is a probability model for the experiment The outcomes of the experiment are described and probability has been assigned to each outcome Experiment 1 has four possible OUTCOMES namely 1 A freshman is drawn 2 A sophomore is drawn 3 Ajunior is drawn 4 A senior is drawn The set of possible outcomes is called the outcome space or the SAMPLE SPACE of the Experiment The sample space in Experiment 1 consists of the four outcomes listed above The outcomes are also referred to as the ELEMENTS of the sample space An EVENT is a collection of elements of the sample space or a subset of the sample space Getting an upperclassman in the above experiment is an event Obtaining a freshman is an event sometimes referred to as an ELEMENTARY EVENT Obtaining a person who is not a freshman is also an event The sample space is an event Since events are identified as collections of points they are what we view as sets As you know we can talk about operations on sets such as union intersection and complement Remember that the union of two sets denoted by U is the set of elements that belongs to one or the other set or both The intersection is the set of elements that belongs to both sets The complement of a set relative to the sample space is the set of objects in the sample space which do not belong to the set Two events are MUTUALLY EXCLUSIVE if they cannot occur simultaneously Freshmen and sophomores are mutually exclusive events in the present example 32 THE AXIOMS FOR ASSIGNING PROBABILITY How should one assign probability to an event This is not quite as innocent a question as it might seem Indeed it turns out to be a question that leads to fundamental issues in epistemology Obviously the assignment cannot be completely arbitrary We begin by giving some rules or axioms which seem reasonable and which seem to impose a reasonable consistency requirement Rule 1 The probability ofthe sample space S is one That is PS 1 This requires that some outcome in the sample space will occur If this were not the case the experiment would not have been described properly Rule 2 For any event E in the sample space its probability is a number between zero and one inclusive That is 0 lt PE lt 1 This rule says that probability is a proportion Rule 3 If E1 E 2 E 3 are a sequence of mutually exclusive events then the probability of E1 or E 2 or E 3 occurring is the sum of the probabilities of individual events occurring For example since the events freshman and sophomore are mutually exclusive in Experiment 1 the probability of the event freshman or sophomore should be the sum of the probability of a freshman and the probability of a sophomore Let us go back to Experiment 1 and see if our rules for probability assignment are satisfied Let us introduce some notation Let E 1 be the event that a freshman is drawn E 2 be the event that a sophomore is drawn E 3 be the event that a junior is drawn E 4 be the event that a senior is drawn There the assignment of probability is as follows PEl1 PEz25 PE345 PEz2 Does the assignment of probability that is given above for Experiment 1 satisfy the three rules Yes The probabilities are all proportions Since the events are mutually exclusive their union is the sample space and the probabilities add to one then the probability of the sample space is one Is the above assignment of probability for Experiment 1 the only one which is consistent with the three Rules The answer is no There are many assignments of probability that satisfy the three Rules Let us give two such examples PE1 13032 13033 13034 25 This assignment satis es the rules The probabilities are all fractions and the probability of the sample space is one A second assignment is PE12 PE28 PE30 PE40 Again one nds that this assignment of probability satis es the three rules So what is going on here Why do people reject the assignments in Examples 2 and 3 in favor of the original assignment Because they don t re ect their beliefs about the chances of the events occurring You may ask whether it is possible to obtain a unique assignment of probability The answer is yes for any given individual One uses a device called a simple lottery Its use in the current example is almost obvious We have an urn that consists of 100 identical balls numbered one through onehundred We also have 100 tickets numbered one through onehundred One ball will be drawn at random from the urn The person holding the ticket with the same number wins the prize If an individual holds four tickets what is the probability of his winning It is 4100 In general the probability of one winning is the ratio of the number of tickets one holds to onehundred So what one has here is an experiment where everyone is agreed on the probability assignment How can this be used to elicit an individuals assignment in Experiment 1 or other experiments What is done is to offer the individual the same prize if a freshman occurs Then one nds out how many lottery tickets are needed to make the person indifferent between participating in the simple lottery and drawing of the person at random from the class 33 SOME IMPLICATIONS OF THE AXIOMS One of the surprising facts is that it is possible to build quite an elaborate system of reasoning about uncertainty on the basis of the three rules given above for probability assignment We now explore some of the implications of the rules Our third rule tells one how to compute the probability of an event which is the union of two events with known probabilities and where the two events are disjoint or mutually exclusive How might one compute the probability of one or the other event occurring if the two events are not mutually exclusive Let A be the event that the person chosen is either a sophomore or junior and let B be the event that the person chosen is an upperclassman Then PA U B PA PB PA and B The logic of this formula is straightforward Note that event A includes juniors but so also does the event B If one is to avoid double counting one has to subtract the probability of getting a junior PA and B from PA PB This formula is called the General Addition Rule Also we are often interested in the probability of an event not occurring Pnot A l PA This is called the probability of the complement It is often useful in calculating probabilities In order to develop further our understanding of Probability we consider Experiment 11 which is an extension of Experiment I EXPERIMENT II A class consists of 100 persons The following table classifies them by class or year and whether or not the person is a business major Business NonBusiness Maj or Maj or Freshman 5 5 Sophomore 15 10 Junior 25 20 Senior 15 5 A person is chosen at random and both year and major are recorded We consider the following probability assignment for Experiment 11 B 1 2 E1 05 05 E2 15 10 E3 25 20 E 15 05 Note that the elements of the sample space are intersections of events For example the probability that the person drawn is both a freshman E1 and a business major B1 is 5100 This is written PEl and B1 05 This is called a JOINT PROBABILITY You should note that the assignment of probability satis es the three rules Suppose one is interested in the probability of getting a sophomore That is one wants PEz How can this probability be obtained from the joint probabilities in Table Note that there are two kinds of sophomores namely business majors and nonbusiness majors These are mutually exclusive categories PEz PE2 and B1 or E2 and B2 PE2 and B1 PE2 and B2 15 10 25 The probability of obtaining a sophomore is an example of a MARGINAL PROBABILITY The term marginal describes the fact that we could sum the probability in the relevant row to get the probability of interest In fact we can get the marginal probabilities for class and major as follows An idea of considerable importance is CONDITIONAL PROBABILITY Conditional probability gives the probability of an event occurring given that a second event occurs For example one may be interested in the probability of drawing a sophomore given that the person drawn is a nonbusiness major In this case one is reducing or conditioning the sample space to nonbusiness majors ten of whom are sophomores Thus the conditional probability of a sophomore given that one draws a nonbusiness major is 1040 It is written as PElez 25 The same answer is obtained if one divides the joint probability of getting a sophomore and a nonbusiness major PE2 and B2 by the marginal of getting a nonbusiness major PB2 PE2 and B2 1 PBz 4 PEz and B2PB2 14 25 This might provide a means of de ning condition probability If A and B are two events in a sample space and if PB gt 0 then the conditional probability of event A given event B is de ned as PAlB PA and BPB PB gt 0 For example the probability of a junior given a business major is PE3lB1 PE3 and B1PB1 256 417 One could also ask what is the conditional probability of a business major given a junior That is 1303le3 PBl and E3PE3 2545 556 Note that the conditional probabilities for year conditioned on major satisfy the three rules Note that PEllBl 0560 083 PElel 1560 25 PE3lB1 2560 417 PE4lB1 1560 25 satisfy the three rules What we have done so far is to start with joint probabilities for the experiment and obtain marginal and conditional probabilities In some practical problems one knows the marginal and conditional probabilities but is interested in the joint probabilities For example if one is told that 60 of the students are business majors and that 083 in the proportion of freshmen among the business majors what is the probability of obtaining a person who is a freshman and a business major One can use the formula for conditional probability to obtain the following general multiplication rule PA and B PAlBPB Applying the result to our example gives PEl and B1 PEllB1PB1 0836 0495 05 One of the issues often faced by decision makers is whether or not two quantities are related or independent We say that two events are INDEPENDENT if the conditional probability is equal to the marginal probability That is PAlB PA With independence the general multiplication rule reduces to the special multiplication rule PA and B PAPB This can also be used as a criterion for independence Is the event of obtaining a junior independent of the event of obtaining a business major Is PE3lB1 PE3 or is PE3 and B1 PE3PB1 Note PE3lB1 417 but PE 45 So they are not independent How about an example of independent events Consider the ip of two coins Is the obtaining of a head on the toss of the second coin independent of the event of obtaining a head on the rst The answer is yes Let us look more closely at Experiment 11 Is the probability of obtaining a sophomore EC independent of the event of obtaining a business major B1 Proceeding formally PEzBl PEz and B1PB1 1560 25 But PE2 PE2B1 PE2Bz 25 So then two events E2 and B1 are independent One way of understanding what is going on here is to recognize that the proportion of sophomores is the same among business majors and nonbusiness majors So the chances a sophomore do not depend on whether you sample from the subpopulation of business majors or nonbusiness majors On the other hand the proportion of juniors among business majors is 2560 467 while the proportion of juniors among nonbusiness majors is 245 Thus the chances of getting a junior depends on whether we sample the business majors or the nonbusiness majors Exercises for Chapter 3 1 John Jones runs a men39s clothing store Twenty percent of those who come into the store purchase a suit thirty per cent purchase shirts and fteen percent purchase both What is the probability that a customer will purchase a shirt or a suit What is the probability that a person will buy shirts given that he buys a suit What is the probability that the customer will order neither shirts nor a suit 2 A grocery store sells both food and alcoholic beverages Sixty percent of customers buy food while twenty percent buy alcoholic beverages Twenty percent buy both What is the probability that a customer buys both food and alcoholic beverages What is the probability that a customer will purchase food or alcoholic beverages What is the probability that he will purchase neither What is the probability that a customer will purchase alcoholic beverages given that he purchases food 3 A rm has three independent devices in its safety mechanism for detecting dangerous levels of a gas The first device is 95 percent effective the second is 99 percent effective while the third is 999 percent effective in detecting a dangerous level of the gas What is the probability that the system will fail to detect a dangerous level of the gas How would the addition of a device that was 80 effective affect the probability of the system detecting dangerous levels of the gas 4 At a computer mail order store fty percent of those who make inquiries by phone order a computer 30 percent order a printer and twenty percent order both What is the probability that a caller will order a computer or a printer What is the probability that the person will order a printer given that he orders a computer What is the probability that he will order neither a computer nor a printer 5 The following table gives a manager s assessment of the probabilities of level of pro tability of a project under different economic conditions State of Economy Recession Normal Boom Low 01 01 005 Level of Pro t Medium 005 03 005 High 005 01 02 Calculate the marginal probabilities for states of the economy and levels of pro t Calculate the conditional probabilities for pro t levels given that the economy suffers a recession Calculate the conditional probabilities for pro t levels given that the economy experiences a boom Is pro t level independent of the state of the economy Explain 6 Ms Y run a coffee and doughnut shop She has found that the key to breaking even on a given day is having at least 100 customers The number of customers seems to depend on weather She has noted that when temperature is above 70 degrees the days are pro table 80 per cent of the time When temperature is less than 70 degrees the days are pro table only 30 percent of the time In June 40 percent of the days have highs above 70 degrees Given that Ms Y had a pro table day on June 10 what is the probability that the temperature was above 70 degrees 7 There are 100 students in a general education class and they have the following characteristics Business NonBusiness Maj or Maj or In State Resident 3 5 3 5 Outof State Resident 15 15 A student is selected at random from the class In the following nd the probability of the event given a The student is a Business major or a NonBusiness major b The student is a Business major and an instate resident c An instate resident given that he is a Business major d A Business major given that he is an instate resident
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