5/2/20131STA291Statistical MethodsLecture 5Disjoint SetsLet A and B denote two events.Disjoint (mutually exclusive) events: A and B are said to be disjoint if there are no outcomes common to both A and B.The notation for this is written as Note: The last symbol denotes the null set or the empty set.AB SA BProbabilities of EventsLet A be the event A = {o1, o2, …, ok}, where o1, o2, …, okare k different outcomes. ThenP(A)=P(o1)+P(o2)++P(ok)Problem: The number on a license plate is any digit between 0 and 9. What is the probability that the first digit is a 3? What is the probability that the first digit is less than 4?35/2/20132Conditional Probability & the Multiplication RuleNote: P(A|B) is read as “the probability that Aoccurs given that B has occurred.”Multiplied out, this gives the multiplication rule:0 provided ,| BPBPBAPBAPBAPBPBAP |4Multiplication Rule Exampleo The multiplication rule:o Example: A disease which occurs in .001 of the population is tested using a method with a false-positive rate of .05 and a false-negative rate of .05. If someone is selected and tested at random, what is the probability they are positive, and the method shows it?BAPBPBAP |5IndependenceIf events A and B are independent, then the events A and B have no influence on each other.So, the probability of A is unaffected by whether B has occurred.Mathematically, if A is independent of B, we write: P(A|B) = P(A)65/2/20133Multiplication Rule and Independent EventsMultiplication Rule for Independent Events: Let A and Bbe two independent events, thenP(AB)=P(A)P(B).Examples:o Flip a coin twice. What is the probability of observing two heads?o Flip a coin twice. What is the probability of getting a head and then a tail? A tail and then a head? One head?o Three computers are ordered. If the probability of getting a “working” computer is 0.7, what is the probability that all three are “working” ?7Terminologyo P(A B) = P(A,B) joint probability of Aand B (of the intersection of A and B)o P(A|B) conditional probability of A given Bo P(A) marginal, or unconditional probability of A8Looking backo Disjoint sets and their probabilitieso Conditional probabilityo Multiplication rule in the general case and for independent
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