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event A with probability P(A). If we obtain new information and learn that a related event, denoted B, already occurred, we will want to take advantage of this information by calculating a newprobability for event A. This new probability is called a conditional probability. Probability of one event given that another event takes place. Denoted P(A¿B) and reads “probabilityof A given B”P(A|B) = P(A∩ B)P(B)or P(B|A) = P(A∩ B)P(A). Ex.Of the population aged 16-21 and not in college, 13.5% are unemployed, 29.05% are high school dropouts and 5.32% are unemployed high school dropouts. What is the probability that a member of this population is unemployed, given that the person is a high school dropout?A = Unemployed, B= High school dropout P(A|B) = P(A ∩ B)P(B)P(A) = .135, P(B) = .2905, P(A ∩B) = .0532 P(A|B) = .0532.2905=.183 or 18.3%-Independent Events:If the probability of event A is not changed by the existence of event B, then events A and B are independent events. The events are NOT related to each other. Two events are independent if: P(A|B) = P(A) orP(B|A) = P(B). Ex. Based on past data, the probability that a customer will order a dessert is 8%. The probability that a customer will order a bottled
beverage is 14% and the probability that a customer will order both is 1.12%. Is ordering a dessert independent of ordering a bottled beverage? P(A) = .08, P(B) = .14, P(A ∩B) = .0112 P(A|B) = P(A ∩ B)P(B)= .0112.14= .08 So, events A ad B are independent.-Multiplication Law: Used to compute probability of the intersection of two events. P(A ∩B) = P(B)P(A|B) orP(A ∩B)=P(A)P(B|A). -Multiplication Law for independent events: P(A ∩B) = P(A)P(B). -Application of the multiplication law for independent events: The probability of n independent events occurring simultaneously is P(A1A2... An) = P(A1) P(A2) ... P(An). Ex. To illustrate system reliability, supposea website has two independent file servers. Each server has 99% reliability. What is the total system reliability? F1= Server 1 failsF2= Server 2 fails P(F1) = .01, P(F2) = .01, P(F1∩F2) = (.01)(.01) = .0001 1 – .0001 =.9999 -Contingency Table is a cross-tabulation of frequencies of two categorical variables into rows and columns. A contingency table with r rows and c columns has rc cells and is called an r X c table.-Marginal Probabilities: The marginal probability of a single event is found by dividing the event’s row or column total by the total sample size.-Joint Probabilities: A joint probability represents the intersection of two events in a contingency table. It is found by dividing the cell at the intersection of the two eventsby the total sample size.-Union of two events:It is found by adding the marginal probability of each event and subtracting the joint probability of the two events.-Conditional Probabilities: It is found by dividing the joint probability of the two events by the row or column total of the conditional event. -Bayes’ Theorem (Two-event case) (Form of conditional probability): Ex. A company requires all job applicants to submit a test for illegal drugs. If the applicant has used illegal drugs, the test has a 90% chance of a positive result. If the applicanthas not used drugs, the test has an 85% chance of a negative result. Actually, 4% of the job applicants have used illegal drugs. If an applicant has a positive test result, what is the probability