# 5a-1. probability (part 1) random experiments random experiments probability rules of probability...

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• 5A-*

• 5A-*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. Sample SpaceRandom Experiments

• 5A-*For example, when CitiBank makes a consumer loan, the sample space is:S = {default, no default}The sample space describing a Wal-Mart customers payment method is:S = {cash, debit card, credit card, check} Sample SpaceRandom Experiments

• 5A-*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: Sample SpaceRandom Experiments

• 5A-*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 levelsIt would be impractical to enumerate this sample space.There are: 2 x 21 x 6 x 4 = 1008 possible outcomes Sample SpaceRandom Experiments

• 5A-*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 beS = {all X such that X > 0}The sample space to describe a randomly chosen students GPA would beS = {X | 0.00 < X < 4.00}or written as S = {X | X > 0} Sample SpaceRandom Experiments

• 5A-*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} EventsRandom Experiments

• 5A-*What are the chances of observing a H or T?These two elementary events are equally likely.S = {H, T}When you buy a lottery ticket, the sample space S = {win, lose} has only two events. EventsRandom ExperimentsAre these two events equally likely to occur?

• 5A-*For example, in a sample space of 6 simple events, we could define the compound eventsA compound event consists of two or more simple events. These are displayed in a Venn diagram:A = {E1, E2}B = {E3, E5, E6} EventsRandom Experiments

• 5A-*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)} EventsRandom Experiments

• 5A-*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 DefinitionsProbability

• 5A-*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 byP(S) = P(E1) + P(E2) + + P(En) = 1 DefinitionsProbability

credit card: 32%debit card: 20%cash: 35%check: 18%Sum = 100%

P(credit card) =.32P(debit card) =.20P(cash) =.35P(check) =.18Sum =1.0

• 5A-*Businesses want to be able to quantify the uncertainty of future events.For example, what are the chances that next months revenue will exceed last years 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

• 5A-*Three approaches to probability: What is Probability?Probability

ApproachExampleEmpiricalThere is a 2 percent chance of twins in a randomly- chosen birth.

ClassicalThere is a 50 % probability of heads on a coin flip.

SubjectiveThere is a 75 % chance that England will adopt the Euro currency by 2010.

• 5A-*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 Empirical ApproachProbability

• 5A-*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. Empirical ApproachProbability

• 5A-*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. 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.). Law of Large NumbersProbabilityConsider the results of 10, 20, 50, and 500 coin flips.

• 5A-*Probability

• 5A-*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 Practical Issues for ActuariesProbability

• 5A-*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. Classical ApproachProbability

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

• 5A-*A subjective probability reflects someones 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? Subjective ApproachProbability

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

• 5A-*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 EventRules of Probability

• 5A-*Since A and A together comprise the entire sample space, P(A) + P(A ) = 1The 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% Complement of an EventRules of Probability

• 5A-*The odds in favor of event A occurring isOdds 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? Odds of an EventRules of Probability

• 5A-*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. Odds of an EventRules of Probability

• 5A-*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 Odds of an EventRules of Probability

• 5A-*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 EventsRules of Probability

• 5A-*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).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 EventsRules of Probability

• 5A-*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 EventsRules of Probability

• 5A-*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?For example, randomly choose a card from a deck of 52 playing cards. Intersection of Two EventsRules of Probability

• 5A-*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.AB General Law of AdditionRules of Probability

• 5A-*For the card example:P(Q) = 4/52 (4 queens in a deck)= 4/52 + 26/52 2/52P(Q R) = P(Q) + P(R) P(Q Q)Q 4/52R 26/52 General Law of AdditionRules 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-*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) = 0In the case of mutually exclusive events, the addition law reduces to:P(A B) = P(A) + P(B) Mutually Exclusive EventsRules of Pr