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Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment.

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## Chapter Two Probability

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**Probability DefinitionsExperiment: Process that generates**observations.Sample Space: Set of all possible outcomes of an experiment.**Event Definitions Event: Subset of outcomes contained in**the sample space.Simple Event: Consists of exactly one outcome.Compound Event:Consists of more than one outcome.**Set Notation ReviewFor Two Events A and B:Union: “A or**B” = A BIntersection:“A and B” = A BComplement: A´Mutually Exclusive: No outcomes in common**Probabilistic Models1) Equally Likely: ØBased on**DefinitionØGames of Chance2) Relative FrequencyØObjective InterpretationØBased on Empirical Data3) Personal ProbabilityØSubjective InterpretationØBased on Degree of Belief**Properties of ProbabilityFor any Event A:P(A) = 1 –**P(A)If A and B are Mutually Exclusive, P(A B) = 0For any two events A and B:P(A B) = P(A) + P(B) – P(A B)**Counting TechniquesProduct Rule for Ordered PairsTree**DiagramsGeneral Product Rule PermutationsCombinations**PermutationAn “ordered” arrangement ofkdistinct objects**taken from a set of n distinct objects.The number of ways of ordering ndistinct objects taken k at a time is Pk,nPk,n = n! / (n-k)!**CombinationAn “unordered” arrangement of k distinct**objects taken from a set of n distinct objects.The number of ways of ordering ndistinct objects taken kat a time is Ck,nCk,n = (nk) = n! / k!(n-k)!**Example:Twenty Five tickets are sold in a lottery, with the**first, second, and third prizes to be determined by a random drawing. Find the number of different ways of drawing the three winning tickets.**Example:Twenty tickets are sold in a lottery, with 5 round**trips to game 1 of the World Series to be determined by a random drawing. Find the number of different ways of drawing the five winning tickets.**Example: A solar system contains 6 Earth-like planets & 4**Gas Giant-like planets. How many ways may we explore this solar system if our resources allow us to only probe 3 Gas Giants and 3 Earth-like planets?**Example:There are 50 students in ISE 261. What is the**probability that at least 2 students have the same birthday? (Ignore leap years).**ExampleA dispute has risen in Watson Engineering concerning**the alleged unequal distribution of 10 computers to three different engineering labs. The first lab (considered to be abominable) required 4 computers; the second lab and third lab needed 3 each. The dispute arose over an alleged ISE 261 random distribution of the computers to the labs which placed all 4 of the fastest computers to the first lab. The Dean desires to known the number of ways of assigning the 10 computers to the three labs before deciding on a course of action. What is the Dean’s next question?**Conditional ProbabilityFor any two events A and B with P(B)**> 0, the conditional probability of A given that B has occurred is defined by:P(A|B) = P(A B)/P(B)**Multiplication RuleFour students have responded to a request**by a blood bank. Blood types of each student are unknown. Blood type A+ is only needed. Assuming one student has this blood type; what is the probability that at least 3 students must be typed to obtain A+?**Conditional ProbabilityExperiment = One toss of a coin.If**the coin is Heads; one die is thrown. Record Number.If the coin is Tails; two die are thrown. Record Sum. What is the Probability that the recorded number will equal 2?**Conditional Probability Problem:30% of interstate highway**accidents involve alcohol use by at least one driver (Event A). If alcohol is involved there is a 60% chance that excessive speed (Event S) is also involved; otherwise, this probability is only 10%. An accident occurs involving speeding! What is the probability that alcohol is involved?P(A) = .30 P(SA|A) = .60 P(A’)= .70 P(SA’|A’)= .10**Bayes’ TheoremA1,A2,….,Ak a collection of k mutually**exclusive and exhaustive events with P(Ai) > 0 for i = 1,…,k. For any other event B for which P(B) > 0: P(Ap|B) = P (Ap B) / P(B) =P(B|Ap) P(Ap) P(B|Ai) P(Ai)**Example: Bayes’ TheoremThe probabilities are equal that**any of 3 urns A1, A2,& A3 will be selected. Given an urn has been selected & the drawn ball is black; what is the probability that the selected urn was A3? A1 contains: 4 W & 1 Black A2 contains: 3 W & 2 Black A3 contains: 1 W & 4 Black**IndependenceTwo events A and B are independent if: P(A|B) =**P(A)OrP(B|A) = P(B)Or P(A B) = P(A) P(B)and are dependent otherwise.**Independence Example:Three brands of coffee, X, Y,& Z are to**be ranked according to taste by a judge. Define the following events as: A: Brand X is preferred to Y B: Brand X is ranked Best C: Brand X is ranked Second D: Brand X is ranked ThirdIf the judge actually has no taste preference & thus randomly assigns ranks to the brands, is event A independent of events B, C, & D?**IndependenceConsider the following 3 events in the toss of a**single die:A: Observe an odd numberB: Observe an even numberC: Observe an 1 or 2Are A & B independent events?Are A & C independent events?**Example:A space probe to Mars has 35 electrical components**in series. If the mission is to have a reliability (probability of success) of 0.90 & if all parts have the same reliability, what is the required reliability of each part?

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