PROBABILITY AND STATISTICS

1. PROBABILITY AND STATISTICS WEEK 3 Onur Doğan

2. Conditional Probability A major use of probability in statistical inference is the updating of probabilitieswhen certain events are observed. The updated probability of event A after welearn that event B has occurred is the conditional probability of A given B. Onur Doğan

3. Conditional Probability Onur Doğan

4. Conditional Probability Suppose that of all individuals buying a certain digitalcamera, 60% include an optionalmemory card in their purchase, 40% include an extra battery, and 30% include both acard and battery. Consider randomly selecting a buyer and let A {memory card purchased}and B {battery purchased}. • What’stheprobabilitythat selected individual purchased an extrabattery is alsopurchased an optional card? • What’stheprobabilitythat selected individual purchased optional card is alsopurchased an extrabattery? Onur Doğan

5. Conditional Probability Rolling Dice. Suppose that two dice were rolled and it was observed that the sum T ofthe two numbers was odd.We shall determine the probability that T was less than 8. Onur Doğan

6. Conditional Probability A news magazine publishes three columns entitled “Art” (A), “Books” (B), and“Cinema” (C). Reading habits of a randomly selected reader with respect to thesecolumns are; • What’s the probability that a selected reader, who reads books also reads arts? • If it is known that selected person reads at least one column, what’s the probability that he reads art? • If it is known that the selected person reads cinema, what’s the probability that he reads book and art also? Onur Doğan

7. Conditional Probability Lottery Ticket. Consider a state lottery game in which six numbers are drawn withoutreplacement from a bin containing the numbers 1–30. Each player tries to match theset of six numbers that will be drawn without regard to the order in which the numbersare drawn. Suppose that you hold a ticket in such a lottery with the numbers 1, 14,15, 20, 23, and 27. You turn on your television to watch the drawing but all you see isone number, 15, being drawn when the power suddenly goes off in your house. Youdon’t even know whether 15 was the first, last, or some in-between draw. However,now that you know that 15 appears in the winning draw, the probability that yourticket is a winner must be higher than it was before you saw the draw. How do youcalculate the revised probability? Onur Doğan

8. Independence • Two events A and B are stochastically independent if the occurrence of A does not affect the probability of B. In other words, two events A and Bare independent if and only if; • P(A /B) = P(A) • P(B / A) = P(B) • P(A  B) = P(A).P(B) Onur Doğan

9. Example It is known that in a laboratory 50 computers have some kind of virus and 50 computers have not. And a virus program have been operated; • Find the probability that virus test is positive when a computer has virus? • Find the probability that virus test is positive when a computer has no virus? • Check if events A and B are independent. Onur Doğan

10. The Multiplication Rule Onur Doğan

11. Example Selecting Two Balls. Suppose that two balls are to be selected at random, withoutreplacement, from a box containing r red balls and b blue balls. We shall determinethe probability p that the first ball will be red and the second ball will be blue. Onur Doğan

12. The Multiplication Rule forConditionalProbabilities Onur Doğan

13. The Multiplication Rule Selecting Four Balls. Suppose that four balls are selected one at a time, withoutreplacement, from a box containing r red balls and b blue balls (r ≥ 2, b ≥ 2). Weshall determine the probability of obtaining the sequence of outcomes red, blue, red,blue. Onur Doğan

14. Example A chain of video stores sells three different brands of DVD players. Of its DVDplayer sales, 50% are brand 1, 30% are brand 2, and 20% arebrand 3. Each manufacturer offers a 1-year warranty on parts and labor. It is knownthat 25% of brand 1’s DVD players require warranty repair work, whereas the correspondingpercentages for brands 2 and 3 are 20% and 10%, respectively. • What is the probability that a randomly selected purchaser has bought a brand 1DVD player that will need repair while under warranty? • What is the probability that a randomly selected purchaser has a DVD player thatwill need repair while under warranty? • If a customer returns to the store with a DVD player that needs warranty repair work,what is the probability that it is a brand 1 DVD player? A brand 2 DVD player?A brand 3 DVD player? Onur Doğan

15. The Law of Total Probability Onur Doğan

16. Example Two boxes contain long boltsand short bolts. Suppose that one boxcontains 60 long bolts and 40 short bolts, and that the other box contains 10 long boltsand 20 short bolts. Suppose also that one box is selected at random and a bolt is thenselected at random from that box. We would like to determine the probability thatthisbolt is long. Onur Doğan

17. Bayes’ Theorem Onur Doğan

18. Example For the previous question; If we know the selected bolt is long, determine the probability that the bolt come from first box? Onur Doğan

19. Example Three different machines M1, M2, and M3were used for producing some items. Suppose that20 percent of the items were produced by machine M1, 30 percent by machine M2,and 50 percent by machineM3. Suppose further that 1 percent of the items producedby machine M1 are defective, that 2 percent of the items produced by machine M2are defective, and that 3 percent of the items produced by machineM3 are defective.Finally, suppose that one item is selected atrandom from the entire batch and it isfound to be defective.We shall determine the probability that this item was produced by machine M2. Onur Doğan

20. Example Customers are used to evaluate preliminary product designs. In the past, 95%ofhighly successful products, 60% of moderately successful products, and10% ofpoor products received good reviews. In addition, 40% of product designs havebeen highly successful, 35% have been moderately successful, and 25% havebeen poor products. • Find the probability that a product receives a good review. • What is the probability that a new design will be highly successful if it receives a good review? Onur Doğan