5A-1

Probability (Part 1) Random Experiments Random Experiments Probability Rules of Probability Rules of Probability Independent Events Independent Events Chapter 5A5A McGraw-Hill/Irwin© 2008 The McGraw-Hill Companies, Inc. All rights reserved.

5A-3 A random experiment is an observational process whose results cannot be known in advance.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.The set of all outcomes (S) is the sample space for the experiment. A sample space with a countable number of outcomes is discrete.A sample space with a countable number of outcomes is discrete.  Sample Space Random Experiments

5A-4 For example, when CitiBank makes a consumer loan, the sample space is: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:The sample space describing a Wal-Mart customer’s payment method is: S = {cash, debit card, credit card, check}  Sample Space Random Experiments

5A-5 For a single roll of a die, the sample space is: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:When two dice are rolled, the sample space is the following pairs:  Sample Space Random Experiments {(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 =S =S =S =

5A-6 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 levelsConsider 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 It would be impractical to enumerate this sample space.It would be impractical to enumerate this sample space. There are: 2 x 21 x 6 x 4 = 1008 possible outcomes  Sample Space Random Experiments

5A-7 If the outcome is a continuous measurement, the sample space can be described by a rule.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 beFor example, the sample space for the length of a randomly chosen cell phone call would be S = {all X such that X > 0} The sample space to describe a randomly chosen student’s GPA would beThe sample space to describe a randomly chosen student’s GPA would be S = {X | 0.00 < X < 4.00} or written as S = {X | X > 0}  Sample Space Random Experiments

5A-8 An event is any subset of outcomes in the sample space.An event is any subset of outcomes in the sample space. A simple event or elementary event, is a single outcome.A simple event or elementary event, is a single outcome. A discrete sample space S consists of all the simple events (E i ):A discrete sample space S consists of all the simple events (E i ): S = {E 1, E 2, …, E n }  Events Random Experiments

5A-9 What are the chances of observing a H or T?What are the chances of observing a H or T? These two elementary events are equally likely.These two elementary events are equally likely. S = {H, T} Consider the random experiment of tossing a balanced coin. What is the sample space?Consider the random experiment of tossing a balanced coin. What is the sample space? When you buy a lottery ticket, the sample space S = {win, lose} has only two events.When you buy a lottery ticket, the sample space S = {win, lose} has only two events.  Events Random Experiments Are these two events equally likely to occur?Are these two events equally likely to occur?

5A-10 For example, in a sample space of 6 simple events, we could define the compound eventsFor example, in a sample space of 6 simple events, we could define the compound events A compound event consists of two or more simple events.A compound event consists of two or more simple events. These are displayed in a Venn diagram:These are displayed in a Venn diagram: A = {E 1, E 2 } B = {E 3, E 5, E 6 }  Events Random Experiments

5A-11 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)} For example, the compound event A = “rolling a seven” on a roll of two dice consists of 6 simple events:  Events Random Experiments

5A-12 The probability of an event is a number that measures the relative likelihood that the event will occur.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:The probability of event A [denoted P(A)], must lie within the interval from 0 to 1: 0 < P(A) < 1 If P(A) = 0, then the event cannot occur. If P(A) = 1, then the event is certain to occur.  Definitions ProbabilityProbability

5A-13 In a discrete sample space, the probabilities of all simple events must sum to unity:In a discrete sample space, the probabilities of all simple events must sum to unity: For example, if the following number of purchases were made byFor example, if the following number of purchases were made by P(S) = P(E 1 ) + P(E 2 ) + … + P(E n ) = 1 credit card: 32% 32% debit card: 20% 20% cash: 35% 35% check: 18% 18% Sum = 100%  Definitions ProbabilityProbability P(credit card) =.32 P(debit card) =.20 P(cash) =.35 P(check) =.18 Sum = 1.0 Probability

5A-14 Businesses want to be able to quantify the uncertainty of future events.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?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?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.The study of probability helps us understand and quantify the uncertainty surrounding the future. ProbabilityProbability

5A-15 Three approaches to probability:Three approaches to probability: ApproachExample Empirical There is a 2 percent chance of twins in a randomly- chosen birth.  What is Probability? ProbabilityProbabilityClassical There is a 50 % probability of heads on a coin flip. Subjective There is a 75 % chance that England will adopt the Euro currency by 2010.

5A-16 Use the empirical or relative frequency approach to assign probabilities by counting the frequency ( f i ) of observed outcomes defined on the experimental sample space.Use the empirical or relative frequency approach to assign probabilities by counting the frequency ( f i ) of observed outcomes defined on the experimental sample space. For example, to estimate the default rate on student loans:For example, to estimate the default rate on student loans: P(a student defaults) = f /n  Empirical Approach ProbabilityProbability number of defaults number of loans=

5A-17 Necessary when there is no prior knowledge of events.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.As the number of observations (n) increases or the number of times the experiment is performed, the estimate will become more accurate.  Empirical Approach ProbabilityProbability

5A-18 The law of large numbers is an important probability theorem that states that a large sample is preferred to a small one.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.Flip a coin 50 times. We would expect the proportion of heads to be near.50. A large n may be needed to get close to.50.A large n may be needed to get close to.50. However, in a small finite sample, any ratio can be obtained (e.g., 1/3, 7/13, 10/22, 28/50, etc.).However, in a small finite sample, any ratio can be obtained (e.g., 1/3, 7/13, 10/22, 28/50, etc.).  Law of Large Numbers ProbabilityProbability Consider the results of 10, 20, 50, and 500 coin flips.Consider the results of 10, 20, 50, and 500 coin flips.

5A-19 ProbabilityProbability

5A-20 Actuarial science is a high-paying career that involves estimating empirical probabilities.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 99For 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  Practical Issues for Actuaries ProbabilityProbability

5A-21 In this approach, we envision the entire sample space as a collection of equally likely outcomes.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).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 refers to the process of assigning probabilities before the event is observed. a priori probabilities are based on logic, not experience.a priori probabilities are based on logic, not experience.  Classical Approach ProbabilityProbability

5A-22 For example, the two dice experiment has 36 equally likely simple events. The P(7) isFor 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:The probability is obtained a priori using the classical approach as shown in this Venn diagram for 2 dice:  Classical Approach ProbabilityProbability

5A-23 A subjective probability reflects someone’s personal belief about the likelihood of an event.A subjective probability reflects someone’s personal belief about the likelihood of an event. Used when there is no repeatable random experiment.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?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?  Subjective Approach ProbabilityProbability

5A-24 These probabilities rely on personal judgment or expert opinion.These probabilities rely on personal judgment or expert opinion. Judgment is based on experience with similar events and knowledge of the underlying causal processes.Judgment is based on experience with similar events and knowledge of the underlying causal processes.  Subjective Approach ProbabilityProbability

5A-25 The complement of an event A is denoted by A′ and consists of everything in the sample space S except event A.The complement of an event A is denoted by A′ and consists of everything in the sample space S except event A.  Complement of an Event Rules of Probability

5A-26 Since A and A′ together comprise the entire sample space, P(A) + P(A′ ) = 1Since 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)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: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%  Complement of an Event Rules of Probability

5A-27 The odds in favor of event A occurring isThe odds in favor of event A occurring is Odds are used in sports and games of chance.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?For a pair of fair dice, P(7) = 6/36 (or 1/6). What are the odds in favor of rolling a 7?  Odds of an Event Rules of Probability

5A-28 On the average, for every time a 7 is rolled, there will be 5 times that it is not rolled.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.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.The odds are 5 to 1 against rolling a 7.  Odds of an Event Rules of Probability In horse racing and other sports, odds are usually quoted against winning.In horse racing and other sports, odds are usually quoted against winning.

5A-29 If the odds against event A are quoted as b to a, then the implied probability of event A is: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) isFor example, if a race horse has a 4 to 1 odds against winning, the P(win) is P(A) =  Odds of an Event Rules of Probability P(win) = or 20%

5A-30 The union of two events consists of all outcomes in the sample space S that are contained either in event A or in event B or both (denoted A  B or “A or B”).  may be read as “or” since one or the other or both events may occur.  Union of Two Events Rules of Probability

5A-31 For example, randomly choose a card from a deck of 52 playing cards.For example, randomly choose a card from a deck of 52 playing cards. It is the possibility of drawing either a queen (4 ways) or a red card (26 ways) or both (2 ways).It is the possibility of drawing either a queen (4 ways) or a red card (26 ways) or both (2 ways). 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?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?  Union of Two Events Rules of Probability

5A-32 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.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.  Intersection of Two Events Rules of Probability

5A-33 It is the possibility of getting both a queen and a red card (2 ways).It is the possibility of getting both a queen and a red card (2 ways). 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?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? For example, randomly choose a card from a deck of 52 playing cards.For example, randomly choose a card from a deck of 52 playing cards.  Intersection of Two Events Rules of Probability

5A-34 The general law of addition states that the probability of the union of two events A and B is: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 A and B  General Law of Addition Rules of Probability

5A-35 For the card example:For the card example: P(Q) = 4/52 (4 queens in a deck) = 4/ /52 – 2/52 P(Q  R) = P(Q) + P(R) – P(Q  Q) Q 4/52 R 26/52 Q and R = 2/52  General Law of Addition Rules of Probability = 28/52 =.5385 or 53.85% P(R) = 26/52 (26 red cards in a deck) P(Q  R) = 2/52 (2 red queens in a deck)

5A-36 Events A and B are mutually exclusive (or disjoint) if their intersection is the null set (  ) that contains no elements.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 In the case of mutually exclusive events, the addition law reduces to:In the case of mutually exclusive events, the addition law reduces to: P(A  B) = P(A) + P(B)  Mutually Exclusive Events Rules of Probability  Special Law of Addition

5A-37 Events are collectively exhaustive if their union is the entire sample space S.Events are collectively exhaustive if their union is the entire sample space S. Two mutually exclusive, collectively exhaustive events are dichotomous (or binary) events.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). Warranty No Warranty  Collectively Exhaustive Events Rules of Probability

5A-38 More than two mutually exclusive, collectively exhaustive events are polytomous events. For example, a Wal-Mart customer can pay by credit card (A), debit card (B), cash (C) or check (D). CreditCard DebitCard Cash Check  Collectively Exhaustive Events Rules of Probability

5A-39 Polytomous events can be made dichotomous (binary) by defining the second category as everything not in the first category.Polytomous events can be made dichotomous (binary) by defining the second category as everything not in the first category. Polytomous Events Binary (Dichotomous) Variable Vehicle type (SUV, sedan, truck, motorcycle) X = 1 if SUV, 0 otherwise  Forced Dichotomy Rules of Probability A randomly-chosen NBA player’s height X = 1 if height exceeds 7 feet, 0 otherwise Tax return type (single, married filing jointly, married filing separately, head of household, qualifying widower) X = 1 if single, 0 otherwise

5A-40 The probability of event A given that event B has occurred.The probability of event A given that event B has occurred. Denoted P(A | B). The vertical line “ | ” is read as “given.”Denoted P(A | B). The vertical line “ | ” is read as “given.” for P(B) > 0 and undefined otherwise  Conditional Probability Rules of Probability

5A-41 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).  Conditional Probability Rules of Probability

5A-42 Of the population aged 16 – 21 and not in college:Of the population aged 16 – 21 and not in college: Unemployed13.5% High school dropouts 29.05% Unemployed high school dropouts 5.32% 5.32% What is the conditional probability that a member of this population is unemployed, given that the person is a high school dropout?What is the conditional probability that a member of this population is unemployed, given that the person is a high school dropout?  Example: High School Dropouts Rules of Probability

5A-43 First defineFirst define U = the event that the person is unemployed D = the event that the person is a high school dropout P(U) =.1350 P(D) =.2905 P(U  D) =.0532 or 18.31% or 18.31% P(U | D) =.1831 > P(U) =.1350P(U | D) =.1831 > P(U) =.1350 Therefore, being a high school dropout is related to being unemployed.Therefore, being a high school dropout is related to being unemployed.  Example: High School Dropouts Rules of Probability

5A-44 Event A is independent of event B if the conditional probability P(A | B) is the same as the marginal probability P(A).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: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: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) P(B) P(B) Independent Events

5A-45 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.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)?What is P(A | B)? Independent Events  Example: Television Ads.3333 or 33%.3333 or 33%

5A-46 So, P(ad A) =.25 P(ad B) =.15 P(A  B) =.05 P(A | B) =.3333So, P(ad A) =.25 P(ad B) =.15 P(A  B) =.05 P(A | B) =.3333 P(A | B) =.3333 ≠ P(A) =.25P(A | B) =.3333 ≠ P(A) =.25 P(A)P(B)=(.25)(.15)=.0375 ≠ P(A  B)=.05P(A)P(B)=(.25)(.15)=.0375 ≠ P(A  B)=.05 Are events A and B independent?Are events A and B independent? Independent Events  Example: Television Ads

5A-47 When P(A) ≠ P(A | B), then events A and B are dependent.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.For dependent events, knowing that event B has occurred will affect the probability that event A will occur. 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.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  Dependent Events Statistical dependence does not prove causality.Statistical dependence does not prove causality.

5A-48 An actuary studies conditional probabilities empirically, using - accident statistics - mortality tables - insurance claims recordsAn 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.Many businesses rely on actuarial services, so a business student needs to understand the concepts of conditional probability and statistical independence. Independent Events  Actuaries Again

5A-49 The probability of n independent events occurring simultaneously is: 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, P(A 1  A 2 ...  A n ) = P(A 1 ) P(A 2 )... P(A n ) if the events are independent F 1 be the event that server 1 fails F 2 be the event that server 2 fails Independent Events  Multiplication Law for Independent Events

5A-50 Applying the rule of independence: The probability that at least one server is “up” is:The probability that at least one server is “up” is: P(F 1  F 2 ) = P(F 1 ) P(F 2 ) = (.01)(.01) = =.9999 or 99.99% So, the probability that both servers are down is.0001.So, the probability that both servers are down is Independent Events  Multiplication Law for Independent Events

5A-51 Redundancy can increase system reliability even when individual component reliability is low.Redundancy can increase system reliability even when individual component reliability is low. NASA space shuttle has three independent flight computers (triple redundancy).NASA space shuttle has three independent flight computers (triple redundancy). Each has an unacceptable.03 chance of failure (3 failures in 100 missions).Each has an unacceptable.03 chance of failure (3 failures in 100 missions). Let F j = event that computer j fails.Let F j = event that computer j fails. Independent Events  Example: Space Shuttle

5A-52 What is the probability that all three flight computers will fail?What is the probability that all three flight computers will fail? P(all 3 fail) = P(F 1  F 2  F 3 ) = or 27 in 1,000,000 missions. = P(F 1 ) P(F 2 ) P(F 3 )  presuming that failures are independent = (0.03)(0.03)(0.03) Independent Events  Example: Space Shuttle

5A-53 How high must reliability be?How high must reliability be? Public carrier-class telecommunications data links are expected to be available % of the time.Public carrier-class telecommunications data links are expected to be available % of the time. The five nines rule implies only 5 minutes of downtime per year.The five nines rule implies only 5 minutes of downtime per year. This type of reliability is needed in many business situations.This type of reliability is needed in many business situations. Independent Events  The Five Nines Rule

5A-54 For example,For example, Independent Events  The Five Nines Rule

5A-55 Suppose a certain network Web server is up only 94% of the time. What is the probability of it being down?Suppose a certain network Web server is up only 94% of the time. What is the probability of it being down? How many independent servers are needed to ensure that the system is up at least 99.99% of the time (or down only =.0001 or.01% of the time)?How many independent servers are needed to ensure that the system is up at least 99.99% of the time (or down only =.0001 or.01% of the time)? P(down) = 1 – P(up) = 1 –.94 =.06 Independent Events  How Much Redundancy is Needed?

5A-56 So, to achieve a 99.99% up time, 4 redundant servers will be needed.So, to achieve a 99.99% up time, 4 redundant servers will be needed. 2 servers: P(F 1  F 2 ) = (0.06)(0.06) = servers: P(F 1  F 2  F 3 ) = (0.06)(0.06)(0.06) = servers: P(F 1  F 2  F 3  F 4 ) = (0.06)(0.06)(0.06)(0.06) = Independent Events  How Much Redundancy is Needed?