Slide Slide 1 Created by Tom Wegleitner, Centreville, Virginia Edited by Olga Pilipets, San Diego, California Multiplication Rule

Slide Slide 2  It is used when we would like to compute the probability of two events happening in succession. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. P(A and B) = P(event A occurs in a first trial and event B occurs in a second trial)

Slide Slide 3  You roll a die two times.  What is the probability that you will get a 5 on the first roll and an even number on the second? Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. P(5)= 1/6 P( even number ) =1/2 P(5 and even number) = 1/6 * 1/2 = 1/12

Slide Slide 4  P(A and B) = P(A) P(B) Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Slide Slide 5  In a Riverhead, New York, case, nine different crime victims listened to voice recordings of five different men. All nine victims identified the same voice as that of the criminal. If the identifications were made by random guesses, find the probability that all nine victims would select the same person. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Slide Slide 6 Independent Events Two events A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other. If A and B are not independent, they are said to be dependent. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Slide Slide 7  Event A: ◦ You are wearing white socks and sandals  Event B: ◦ A girl agrees to go out with you. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Slide Slide 8  Finding that your kitchen toaster is not working  Finding that your refrigerator is not working. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Drinking or using drugs until your driving ability is impaired Being involved in a car crash Finding that your calculator is working Finding that your cell phone is working

Slide Slide 9 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Key Point – Conditional Probability The probability for the second event B should take into account the fact that the first event A has already occurred. NOTATION P(B/A) - the probability of event B occurring after it is assumed that event A has already occurred

Slide Slide 10  Event A: ◦ Drawing a red card from a deck of cards  Event B: ◦ Drawing a second red card form a deck of cards (drawing is made without replacement) Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. P(A) = 26/52 P(B) = 25 / 51

Slide Slide 11  Independent events P(A and B) = P(A) P(B)  Dependent events P(A and B) = P(A) P(B/A) Note that if A and B are independent events, P(B / A) is really the same as P(B). Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Slide Slide 12 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Applying the Multiplication Rule

Slide Slide 13  Use the data in the table, which summarizes results from 985 pedestrian deaths that were caused by accidents (based on data from the National Highway Traffic Safety Administration).  If two different pedestrian deaths are randomly selected, find the probability that they both involve intoxicated pedestrians. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Slide Slide 14 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Slide Slide 15 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Summary of Fundamentals  In the addition rule, the word “or” in P(A or B) suggests addition. Add P(A) and P(B), being careful to add in such a way that every outcome is counted only once.  In the multiplication rule, the word “and” in P(A and B) suggests multiplication. Multiply P(A) and P(B), but be sure that the probability of event B takes into account the previous occurrence of event A.

Slide Slide 16 Created by Tom Wegleitner, Centreville, Virginia Edited by Olga Pilipets, San Diego, California Complements and Conditional Probability

Slide Slide 17 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Key Concept In this section we look at 1.the probability of getting at least one of some specified event; and 2.the concept of conditional probability which is the probability of an event given the additional information that some other event has already occurred.

Slide Slide 18  You are expecting 50 electrocardiograph units to be shipped to you and you were told that all of them are free of defects. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. When 50 electrocardiograph units are shipped, at least one of them is defected. When Jake asks 12 different women for a date, at least one of them accepts. When Jake asks 12 different women for a date, none of them accepts.

Slide Slide 19  The complement of getting at least one item of a particular type is that you get no items of that type. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.  “At least one” is equivalent to “one or more.”

Slide Slide 20  If a couple plans to have 3 children, what is the probability that there will be at least one girl? Assume that both genders are equally likely to be born and the gender of a child is independent of the genders of his/her siblings. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Slide Slide 21  Step 1: Use a symbol to represent the event desired Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Step 2: Identify the event that is the complement of A. Step 3: Find the Probability of the complement. Step 4: Find P(A) by evaluating 1 – P(A)

Slide Slide 22 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Key Principle To find the probability of at least one of something, calculate the probability of none, then subtract that result from 1. That is, P(at least one) = 1 – P(none).

Slide Slide 23 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Definition A conditional probability of an event is a probability that an event B occurs given that event A has already occurred. P(B A) denotes the conditional probability.

Slide Slide 24 Driving a red car? YesNo Males1030 Females525 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. What is the probability that a randomly selected student drives a red car? What is the probability that a randomly selected student is driving a red car given it is a female?

Slide Slide 25 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Formula Conditional probability of event B occurring given that event A has already occurred can be found by dividing the probability of events A and B both occurring by the probability of event A: P(B A) = P(A and B) P(A)

Slide Slide 26 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Recap In this section we have discussed:  Concept of “at least one.”  Conditional probability.  Intuitive approach to conditional probability.

Slide Slide 27  Finish problems on the worksheet Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.