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## Probability (Part 1)

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**5**Chapter Probability (Part 1) Random Experiments Probability Rules of Probability Independent Events**Random Experiments**• Sample Space • A random experiment is an observational process whose results cannot be known in advance. • The set of all outcomes (S) is the sample space for the experiment. • A sample space with a countable number of outcomes is discrete.**Random Experiments**• Sample Space • For example, when CitiBank makes a consumer loan, the sample space is: S = {default, no default} • The sample space describing a Wal-Mart customer’s payment method is: S = {cash, debit card, credit card, check}**{(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2),**(2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)} S = Random Experiments • Sample Space • For a single roll of a die, the sample space is: S = {1, 2, 3, 4, 5, 6} • When two dice are rolled, the sample space is the following pairs:**Random Experiments**• Sample Space • Consider the sample space to describe a randomly chosen United Airlines employee by 2 genders, 21 job classifications, 6 home bases (major hubs) and 4 education levels There are: 2 x 21 x 6 x 4 = 1008 possible outcomes • It would be impractical to enumerate this sample space.**Random Experiments**• Sample Space • If the outcome is a continuous measurement, the sample space can be described by a rule. • For example, the sample space for the length of a randomly chosen cell phone call would be S = {all X such that X > 0} or written as S = {X | X > 0} • The sample space to describe a randomly chosen student’s GPA would be S = {X | 0.00 <X < 4.00}**Random Experiments**• Events • An event is any subset of outcomes in the sample space. • A simple event or elementary event, is a single outcome. • A discrete sample space S consists of all the simple events (Ei): S = {E1, E2, …, En}**Consider the random experiment of tossing a balanced coin.**What is the sample space? Random Experiments • Events S = {H, T} • What are the chances of observing a H or T? • These two elementary events are equally likely. • When you buy a lottery ticket, the sample space S = {win, lose} has only two events. • Are these two events equally likely to occur?**Random Experiments**• Events • A compound event consists of two or more simple events. • For example, in a sample space of 6 simple events, we could define the compound events A = {E1, E2} B = {E3, E5, E6} • These are displayed in a Venn diagram:**For example, the compound event A = “rolling a seven” on**a roll of two dice consists of 6 simple events: Random Experiments • Events • Many different compound events could be defined. • Compound events can be described by a rule. S = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}**If P(A) = 1, then the event is certain to occur.**If P(A) = 0, then the event cannot occur. Probability • Definitions • The probability of an event is a number that measures the relative likelihood that the event will occur. • The probability of event A [denoted P(A)], must lie within the interval from 0 to 1: 0 <P(A) < 1**Probability**Probability • Definitions • In a discrete sample space, the probabilities of all simple events must sum to unity: P(S) = P(E1) + P(E2) + … + P(En) = 1 • For example, if the following number of purchases were made by**Probability**• Businesses want to be able to quantify the uncertainty of future events. • For example, what are the chances that next month’s revenue will exceed last year’s average? • How can we increase the chance of positive future events and decrease the chance of negative future events? • The study of probability helps us understand and quantify the uncertainty surrounding the future.**Probability**• What is Probability? • Three approaches to probability:**number of defaults**number of loans = Probability • Empirical Approach • Use the empirical or relative frequency approach to assign probabilities by counting the frequency (fi) of observed outcomes defined on the experimental sample space. • For example, to estimate the default rate on student loans: P(a student defaults) = f /n**Probability**• Empirical Approach • Necessary when there is no prior knowledge of events. • As the number of observations (n) increases or the number of times the experiment is performed, the estimate will become more accurate.**Probability**• Law of Large Numbers • The law of large numbers is an important probability theorem that states that a large sample is preferred to a small one. • Flip a coin 50 times. We would expect the proportion of heads to be near .50. • However, in a small finite sample, any ratio can be obtained (e.g., 1/3, 7/13, 10/22, 28/50, etc.). • A large n may be needed to get close to .50. • Consider the results of 10, 20, 50, and 500 coin flips.**Probability**• Practical Issues for Actuaries • Actuarial science is a high-paying career that involves estimating empirical probabilities. • For example, actuaries - calculate payout rates on life insurance, pension plans, and health care plans- create tables that guide IRA withdrawal rates for individuals from age 70 to 99**Probability**• Practical Issues for Actuaries • Challenges that actuaries face: - Is n “large enough” to say that f/n has become a good approximation to P(A)? - Was the experiment repeated identically? - Is the underlying process invariant over time? - Do nonstatistical factors override data collection? - What if repeated trials are impossible?**Probability**• Classical Approach • In this approach, we envision the entire sample space as a collection of equally likely outcomes. • Instead of performing the experiment, we can use deduction to determine P(A). • a priori refers to the process of assigning probabilities before the event is observed. • a priori probabilities are based on logic, not experience.**Probability**• Classical Approach • For example, the two dice experiment has 36 equally likely simple events. The P(7) is • The probability is obtained a priori using the classical approach as shown in this Venn diagram for 2 dice:**Probability**• Subjective Approach • A subjective probability reflects someone’s personal belief about the likelihood of an event. • Used when there is no repeatable random experiment. • For example,- What is the probability that a new truck product program will show a return on investment of at least 10 percent?- What is the probability that the price of GM stock will rise within the next 30 days?**Probability**• Subjective Approach • These probabilities rely on personal judgment or expert opinion. • Judgment is based on experience with similar events and knowledge of the underlying causal processes.**Rules of Probability**• Complement of an Event • The complement of an event A is denoted by A′ and consists of everything in the sample space S except event A.**Rules of Probability**• Complement of an Event • Since A and A′ together comprise the entire sample space, P(A) + P(A′ ) = 1 • The probability of A′ is found by P(A′ ) = 1 – P(A) • For example, The Wall Street Journal reports that about 33% of all new small businesses fail within the first 2 years. The probability that a new small business will survive is: P(survival) = 1 – P(failure) = 1 – .33 = .67 or 67%**Rules of Probability**• Odds of an Event • The odds in favor of event A occurring is • Odds are used in sports and games of chance. • For a pair of fair dice, P(7) = 6/36 (or 1/6). What are the odds in favor of rolling a 7?**In horse racing and other sports, odds are usually quoted**against winning. Rules of Probability • Odds of an Event • On the average, for every time a 7 is rolled, there will be 5 times that it is not rolled. • In other words, the odds are 1 to 5 in favor of rolling a 7. • The odds are 5 to 1 against rolling a 7.**P(A) =**or 20% P(win) = Rules of Probability • Odds of an Event • If the odds against event A are quoted as b to a, then the implied probability of event A is: • For example, if a race horse has a 4 to 1 odds against winning, the P(win) is**Rules of Probability**• Union of Two Events • The union of two events consists of all outcomes in the sample space S that are contained either in event Aor in event Bor both (denoted A B or “A or B”). may be read as “or” since one or the other or both events may occur.**Rules of Probability**• Union of Two Events • For example, randomly choose a card from a deck of 52 playing cards. • If Q is the event that we draw a queen and R is the event that we draw a red card, what is Q R? • It is the possibility of drawing either a queen (4 ways) or a red card (26 ways) or both (2 ways).**Rules of Probability**• Intersection of Two Events • The intersection of two events A and B(denoted A B or “A and B”) is the event consisting of all outcomes in the sample space S that are contained in both event A and event B. may be read as “and” since both events occur. This is a joint probability.**Rules of Probability**• Intersection of Two Events • For example, randomly choose a card from a deck of 52 playing cards. • If Q is the event that we draw a queen and R is the event that we draw a red card, what is Q R? • It is the possibility of getting both a queen and a red card (2 ways).**A and B**Rules of Probability • General Law of Addition • The general law of addition states that the probability of the union of two events A and B is: P(A B) = P(A) + P(B) – P(A B) When you add the P(A) and P(B) together, you count the P(A and B) twice. So, you have to subtract P(A B) to avoid over-stating the probability. A B**Q and R = 2/52**Rules of Probability • General Law of Addition • For the card example: P(Q) = 4/52 (4 queens in a deck) P(R) = 26/52 (26 red cards in a deck) P(Q R) = 2/52 (2 red queens in a deck) P(Q R) = P(Q) + P(R) – P(Q Q) = 4/52 + 26/52 – 2/52 = 28/52 = .5385 or 53.85% Q4/52 R26/52**Rules of Probability**• Mutually Exclusive Events • Events A and B are mutually exclusive (or disjoint) if their intersection is the null set () that contains no elements. If A B = , then P(A B) = 0 • Special Law of Addition • In the case of mutually exclusive events, the addition law reduces to: P(A B) = P(A) + P(B)**Rules of Probability**• Collectively Exhaustive Events • Events are collectively exhaustive if their union is the entire sample space S. • Two mutually exclusive, collectively exhaustive events are dichotomous (or binary) events. For example, a car repair is either covered by the warranty (A) or not (B). NoWarranty Warranty**Rules of Probability**• Collectively Exhaustive Events • More than two mutually exclusive, collectively exhaustive events are polytomous events. Cash Debit Card Credit Card Check For example, a Wal-Mart customer can pay by credit card (A), debit card (B), cash (C) or check (D).**Rules of Probability**• Forced Dichotomy • Polytomous events can be made dichotomous (binary) by defining the second category as everything not in the first category.**Rules of Probability**• Conditional Probability • The probability of event Agiven that event B has occurred. • Denoted P(A | B). The vertical line “ | ” is read as “given.” for P(B) > 0 and undefined otherwise**Rules of Probability**• Conditional Probability • Consider the logic of this formula by looking at the Venn diagram. The sample space is restricted to B, an event that has occurred. A B is the part of B that is also in A. The ratio of the relative size of A B to B is P(A | B).**Rules of Probability**• Example: High School Dropouts • Of the population aged 16 – 21 and not in college: • What is the conditional probability that a member of this population is unemployed, given that the person is a high school dropout?**or 18.31%**Rules of Probability • Example: High School Dropouts • First define U = the event that the person is unemployed D = the event that the person is a high school dropout P(D) = .2905 P(UD) = .0532 P(U) = .1350 • P(U | D) = .1831 > P(U) = .1350 • Therefore, being a high school dropout is related to being unemployed.**Independent Events**• Event A is independent of event B if the conditional probability P(A | B) is the same as the marginal probability P(A). • To check for independence, apply this test: If P(A | B) = P(A) then event A is independent of B. • Another way to check for independence: If P(A B) = P(A)P(B) then event A is independent of event B since P(A | B) = P(A B) = P(A)P(B) = P(A) P(B) P(B)**.3333 or 33%**Independent Events • Example: Television Ads • Out of a target audience of 2,000,000, ad A reaches 500,000 viewers, B reaches 300,000 viewers and both ads reach 100,000 viewers. • What is P(A | B)?**Independent Events**• Example: Television Ads • So, P(ad A) = .25P(ad B) = .15P(AB) = .05P(A | B) = .3333 • Are events A and B independent? • P(A | B) = .3333 ≠ P(A) = .25 • P(A)P(B)=(.25)(.15)=.0375 ≠ P(AB)=.05**Independent Events**• Dependent Events • When P(A) ≠ P(A | B), then events A and B are dependent. • For dependent events, knowing that event B has occurred will affect the probability that event A will occur. • Statistical dependence does not prove causality. • For example, knowing a person’s age would affect the probability that the individual uses text messaging but causation would have to be proven in other ways.**Independent Events**• Actuaries Again • An actuary studies conditional probabilities empirically, using - accident statistics - mortality tables - insurance claims records • Many businesses rely on actuarial services, so a business student needs to understand the concepts of conditional probability and statistical independence.**Independent Events**• Multiplication Law for Independent Events • The probability of n independent events occurring simultaneously is: P(A1A2... An) = P(A1) P(A2) ... P(An) if the events are independent • To illustrate system reliability, suppose a Web site has 2 independent file servers. Each server has 99% reliability. What is the total system reliability? Let, F1 be the event that server 1 fails F2 be the event that server 2 fails