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Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. Understanding Probability and Long-Term Expectations Chapter 16
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 2 Probability What does the word probability mean? Two distinct interpretations: For the probability of winning a lottery based on buying a single ticket -- we can quantify the chances exactly. (Relative Frequency interpretation) For the probability that we will eventually buy a home -- we are basing our assessment on personal beliefs about how life will evolve for us.(Personal Probability interpretation)
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 3 The Relative-Frequency Interpretation Examples: Buying a weekly lottery ticket and observing whether it is a winner. Commuting to work daily and observing whether a certain traffic signal is red when we encounter it. Testing individuals in a population and observing whether they carry a gene for a certain disease. Observing births and noting if baby is male or female. Relative-frequency interpretation: applies to events that can be repeated over and over again. The events probability is the proportion of times the event occurs in the long run.
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 4 Idea of Long-Run Relative Frequency Probability = proportion of time it occurs over the long run Long-run relative frequency of males born in the United States is about 0.512. (Information Please Almanac, 1991, p. 815) Possible results for relative frequency of male births: Proportion of male births jumps around at first but starts to settle down just above.51 in the long run.
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 5 Determining the Probability of an Outcome Method 1: Make an Assumption about the Physical World Method 2: Observe the Relative Frequency Example: assume coins made such that they are equally likely to land with heads or tails up when flipped. probability of a flipped coin showing heads up is ½. Example: observe the relative frequency of male births in a given city over the course of a year. In 1987 there were a total of 3,809,394 live births in the U.S., of which 1,951,153 where males. probability of male birth is 1,951,153/3,809,394 = 0.5122
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 6 Summary of Relative-Frequency Interpretation of Probability Can be applied when situation can be repeated numerous times and outcome observed each time. Relative frequency should settle down to constant value over long run, which is the probability. Does not apply to situations where outcome one time is influenced by or influences outcome the next time.
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 7 The Personal-Probability Interpretation Examples: Probability of finding a parking space downtown on Saturday. Probability that a particular candidate for a position would fit the job best. Person probability: the degree to which a given individual believes the event will happen. The personal assessment of an events likelihood.
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 8 Probability Definitions Event: The general situation in which we are making an observation. Outcomes: Collection of all the possible results in the event. Ex. What is the probability of getting a head when we flip a coin? The event would be the coin flip and the possible outcomes would be heads or tails Ex. What is the probability of getting a 2 when we roll a die? The event would be the roll of the die and the possible outcomes would be 1,2,3,4,5 or 6.
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 9 Probability Rules: Our goal is to assign probabilities to all possible outcomes of an event. Denote an outcome as O i and the probability of that event as P(O i ). However, we must assign the probabilities with the following restrictions: 1)Probabilities are NEVER negative. For each outcome of the event we must have that: 2)A probability of 0 means the outcome can never happen 3)A probability of 1 means we are certain that the outcome will happen 4)The total probabilities of all outcomes for an event must add up to one. If there are n outcomes in an event then we have:
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 10 Basic Operations: Conjugate Given an event (which we will call A) the conjugate of that event (denoted as A c ) is the event that A does NOT occur. Event A Event A c Since we know that the probability of us being completely certain about something is 1 we know that Therefore, the probability that an event doesn’t occur is 1 minus the probability that the event does occur.
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 11 Basic Operations: AND Given two events (we will call them event A and event B), we want to know what is the probability that both event A and event B occur. We denote this as. Event A For now we will only know how to calculate the “and” probability when the two events are independent. This means that Event A does not depend on Event B. Event B Therefore, when A and B are independent events we have the following formula:
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 12 Basic Operations: OR Given two events (we will call them event A and event B), we want to know what is the probability that either event A or event B occur. We denote this as. To find the probability of Event A occurring or Event B occurring we add the probability of A and the probability of B, but in doing so we will and the center area twice. Therefore, we must subtract off one of the center areas. When A and B are mutually exclusive events this means that event A and event B can not occur at the same time. Therefore the probability of A and B is zero and our formula reduces to: Event A Event B
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 13 Basic Operations: Subsets Given two events (we will call them event A and event B), event A is a subset of event B if everything in A is also in B. Since we know that probabilities can not be negative we know that the probability of A must be smaller that the probability of B. Event A Event B Event A
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 14 When Will It Happen? Probability an outcome will occur on any given instance is p. Probability the outcome will not occur is (1 – p). Outcome each time is independent of outcome all other times. Probability it doesn’t occur on 1 st try but does occur on 2 nd try is (1 – p)p.
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 15 Example 11: Number of Births to First Girls Probability of a birth resulting in a boy is about.51, and the probability of a birth resulting in a girl is about.49. Suppose couple will continue having children until have a girl. Assuming outcomes of births are independent of each other, probabilities of having the first girl on the first, second, third, fifth, and seventh tries are shown below.
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 16 Accumulated Probability Probability of 1 st occurrence not happening by occasion n is (1 – p) n. Probability 1 st occurrence has happened by occasion n is [1 – (1 – p) n ]. Example 12: Getting Infected with HIV Suppose probability of infected is 1/500 = 0.002. Probability of not infected is 0.998. So probability of infection by 2 nd encounter is [1 – (1 – 0.002) 2 ] = 0.003996.
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 17 Long-Term Gains, Losses, and Expectations Long-Term Outcomes Can Be Predicted Example 14: Insurance Policies A simple case: All customers charged $500/year, 10% submit a claim in any given year and claims always for $1500. How much can the company expect to make per customer? Average Gain =.90 ($500) –.10 ($1000) = $350 per customer
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 18 Expected Value (EV) EV = expected value = A 1 p 1 + A 2 p 2 + A 3 p 3 +... + A k p k where A 1, A 2, A 3, …, A k are the possible amounts and p 1, p 2, p 3, …, p k are the associated probabilities. Example 14, continued: A 1 p 1 + A 2 p 2 = (.10)(−$1000) + (.90)($500) = $350 = expected value of company’s profit per customer. Key: Expected value is the average value per measurement over the long run and not necessarily a typical value for any one occasion or person.
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 19 Expected Value as Mean Number If measurement taken over a large group of individuals, the expected value can be interpreted as the mean value per individual. Example: Suppose 40% of people in a population smoke a pack of cigarettes a day (20 cigarettes) and remaining 60% smoke none. Expected number of cigarettes smoked per day by one person: EV = (0.40)(20 cigarettes) + (0.60)(0 cigarettes) = 8 cigarettes On average 8 cigarettes are smoked per person per day. Here again, the EV is not a value we actually expect to measure on any one individual.
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