Example Suppose that a deck of 52 cards containing four aces is shuffled thoroughly and the cards are then distributed among four players so that each player receives 13 cards. We shall determine the probability that each player will receive one ace. Onur Doğan 2016-2017

Example Suppose that a fair coin is to be tossed 7 times, and it is desired to determine; the probability p of obtaining three heads the probability p of obtaining three or fewer heads. Onur Doğan 2016-2017

Binomial Coefficients Onur Doğan 2016-2017

Multinomial Coefficients Onur Doğan 2016-2017

Multinomial Coefficients Onur Doğan 2016-2017

Example Onur Doğan 2016-2017

PROBABILITY AND STATISTICS WEEK 4 Onur Doğan 2016-2017

Conditional Probability A major use of probability in statistical inference is the updating of probabilities when certain events are observed. The updated probability of event A after we learn that event B has occurred is the conditional probability of A given B. Onur Doğan 2016-2017

Conditional Probability Onur Doğan 2016-2017

Conditional Probability Suppose that of all individuals buying a certain digital camera, 60% include an optional memory card in their purchase, 40% include an extra battery, and 30% include both a card and battery. Consider randomly selecting a buyer and let A {memory card purchased} and B {battery purchased}. What’s the probability that selected individual purchased an extra battery is also purchased an optional card? What’s the probability that selected individual purchased optional card is also purchased an extra battery? Onur Doğan 2016-2017

Conditional Probability Rolling Dice. Suppose that two dice were rolled and it was observed that the sum T of the two numbers was odd. We shall determine the probability that T was less than 8. Onur Doğan 2016-2017

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 these columns 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 2016-2017

Conditional Probability Lottery Ticket. Consider a state lottery game in which six numbers are drawn without replacement from a bin containing the numbers 1–30. Each player tries to match the set of six numbers that will be drawn without regard to the order in which the numbers are 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 is one number, 15, being drawn when the power suddenly goes off in your house. You don’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 your ticket is a winner must be higher than it was before you saw the draw. How do you calculate the revised probability? Onur Doğan 2016-2017

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 B are independent if and only if; P(A /B) = P(A) P(B / A) = P(B) P(A  B) = P(A).P(B) Onur Doğan 2016-2017

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. Test Results Negative Positive (B) Corrupted (A) 10 40 Not corrupted 45 5 Onur Doğan 2016-2017

The Multiplication Rule Onur Doğan 2016-2017

Example Selecting Two Balls. Suppose that two balls are to be selected at random, without replacement, from a box containing r red balls and b blue balls. We shall determine the probability p that the first ball will be red and the second ball will be blue. Onur Doğan 2016-2017

The Multiplication Rule for Conditional Probabilities Onur Doğan 2016-2017

The Multiplication Rule Selecting Four Balls. Suppose that four balls are selected one at a time, without replacement, from a box containing r red balls and b blue balls (r ≥ 2, b ≥ 2). We shall determine the probability of obtaining the sequence of outcomes red, blue, red, blue. Onur Doğan 2016-2017

Example A chain of video stores sells three different brands of DVD players. Of its DVD player sales, 50% are brand 1, 30% are brand 2, and 20% are brand 3. Each manufacturer offers a 1-year warranty on parts and labor. It is known that 25% of brand 1’s DVD players require warranty repair work, whereas the corresponding percentages for brands 2 and 3 are 20% and 10%, respectively. What is the probability that a randomly selected purchaser has bought a brand 1 DVD player that will need repair while under warranty? What is the probability that a randomly selected purchaser has a DVD player that will 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 2016-2017

The Law of Total Probability Onur Doğan 2016-2017

Example Two boxes contain long bolts and short bolts. Suppose that one box contains 60 long bolts and 40 short bolts, and that the other box contains 10 long bolts and 20 short bolts. Suppose also that one box is selected at random and a bolt is then selected at random from that box. We would like to determine the probability that this bolt is long. Onur Doğan 2016-2017

Bayes’ Theorem Onur Doğan 2016-2017

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 2016-2017

Example Three different machines M1, M2, and M3 were used for producing some items. Suppose that 20 percent of the items were produced by machine M1, 30 percent by machine M2, and 50 percent by machine M3. Suppose further that 1 percent of the items produced by machine M1 are defective, that 2 percent of the items produced by machine M2 are defective, and that 3 percent of the items produced by machine M3 are defective. Finally, suppose that one item is selected at random from the entire batch and it is found to be defective. We shall determine the probability that this item was produced by machine M2. Onur Doğan 2016-2017

Example Customers are used to evaluate preliminary product designs. In the past, 95% of highly successful products, 60% of moderately successful products, and 10% of poor products received good reviews. In addition, 40% of product designs have been highly successful, 35% have been moderately successful, and 25% have been 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 2016-2017