Copyright © 2010 Pearson Education, Inc. Chapter 15 Probability Rules!

Copyright © 2010 Pearson Education, Inc. Pizza rollsChips and salsa Cookiestotals Minecraft Pictionary Monopoly Apples to Apples totals Slide If you choose a person at random, what is the probability that that person: Plays minecraft? Eats pizza rolls? Plays minecraft and eats pizza rolls? Plays minecraft or eats pizza rolls? Is a pizza roll eater who plays minecraft? Is a minecraft player who eats pizza rolls?

Copyright © 2010 Pearson Education, Inc. Slide

Copyright © 2010 Pearson Education, Inc. Slide The General Addition Rule When two events A and B are disjoint, we can use the addition rule for disjoint events from Chapter 14: P(A  B) = P(A) + P(B) However, when our events are not disjoint, this earlier addition rule will double count the probability of both A and B occurring. Thus, we need the General Addition Rule. Let’s look at a picture…

Copyright © 2010 Pearson Education, Inc. Slide The General Addition Rule (cont.) General Addition Rule: For any two events A and B, P(A  B) = P(A) + P(B) – P(A  B) The following Venn diagram shows a situation in which we would use the general addition rule:

Copyright © 2010 Pearson Education, Inc. Slide The General Addition Rule (cont.) A survey of college students found that 56% live on campus, 62% are on a campus meal program, and 42% do both. What is the probability that: a student either lives or eats on campus? a student lives off campus and doesn’t have a meal program? a student lives in a dorm but doesn’t have a meal program?

Copyright © 2010 Pearson Education, Inc. Slide It Depends… Back in Chapter 3, we looked at contingency tables and talked about conditional distributions. When we want the probability of an event from a conditional distribution, we write P(B|A) and pronounce it “the probability of B given A.” A probability that takes into account a given condition is called a conditional probability.

Copyright © 2010 Pearson Education, Inc. Slide It Depends… (cont.) To find the probability of the event B given the event A, we restrict our attention to the outcomes in A. We then find the fraction of those outcomes B that also occurred. Note: P(A) cannot equal 0, since we know that A has occurred.

Copyright © 2010 Pearson Education, Inc. Slide It Depends… (cont.) Our survey found that 56% live on campus, 62% have a meal program, 42% do both. While dining in a campus facility open only to students with meal programs you meet someone new. What is the probability they live on campus?

Copyright © 2010 Pearson Education, Inc. Slide The General Multiplication Rule When two events A and B are independent, we can use the multiplication rule for independent events from Chapter 14: P(A  B) = P(A) x P(B) However, when our events are not independent, this earlier multiplication rule does not work. Thus, we need the General Multiplication Rule.

Copyright © 2010 Pearson Education, Inc. Slide The General Multiplication Rule (cont.) We encountered the general multiplication rule in the form of conditional probability. Rearranging the equation in the definition for conditional probability, we get the General Multiplication Rule: For any two events A and B, P(A  B) = P(A)  P(B|A) or P(A  B) = P(B)  P(A|B)

Copyright © 2010 Pearson Education, Inc. Are Monopoly and Chips and Salsa independent? Slide Pizza rollsChips and salsa Cookiestotals Minecraft Pictionary Monopoly Apples to Apples totals

Copyright © 2010 Pearson Education, Inc. Slide Independence Independence of two events means that the outcome of one event does not influence the probability of the other. With our new notation for conditional probabilities, we can now formalize this definition: Events A and B are independent whenever P(B|A) = P(B). (Equivalently, events A and B are independent whenever P(A|B) = P(A).)

Copyright © 2010 Pearson Education, Inc. Slide Independence Our survey found that 56% live on campus, 62% have a meal program, 42% do both. Are living on campus and having a meal plan independent? Are they disjoint? Events A and B are independent whenever P(B|A) = P(B).

Copyright © 2010 Pearson Education, Inc. Slide Independent ≠ Disjoint Disjoint events cannot be independent! Well, why not? Since we know that disjoint events have no outcomes in common, knowing that one occurred means the other didn’t. Thus, the probability of the second occurring changed based on our knowledge that the first occurred. It follows, then, that the two events are not independent. A common error is to treat disjoint events as if they were independent, and apply the Multiplication Rule for independent events—don’t make that mistake.

Copyright © 2010 Pearson Education, Inc. Slide Depending on Independence It’s much easier to think about independent events than to deal with conditional probabilities. It seems that most people’s natural intuition for probabilities breaks down when it comes to conditional probabilities. Don’t fall into this trap: whenever you see probabilities multiplied together, stop and ask whether you think they are really independent.

Copyright © 2010 Pearson Education, Inc. Are “red card” and “spade” independent? Mutually exclusive? Are “red card” and “ace” independent? Mutually exclusive? Are “face card” and “king” independent? Mutually exclusive? Slide

Copyright © 2010 Pearson Education, Inc. Slide Drawing Without Replacement Sampling without replacement means that once one individual is drawn it doesn’t go back into the pool. We often sample without replacement, which doesn’t matter too much when we are dealing with a large population. However, when drawing from a small population, we need to take note and adjust probabilities accordingly. Drawing without replacement is just another instance of working with conditional probabilities.

Copyright © 2010 Pearson Education, Inc. Checking for Independence Students in grades 4, 5, and 6 were surveyed as to what their primary goal was. If we select a student at random, what is P(girl) P(girl ∩ popular) P(grades|boy) Slide GradesPopularSportstotal Boy Girl Total

Copyright © 2010 Pearson Education, Inc. P(sports|girl) P(sports|boy) P(sports) If events are independent, the conditional probability should be the same as the overall probability. Slide

Copyright © 2010 Pearson Education, Inc. Slide What is the probability that a male wears jeans? What is the probability that someone wearing jeans is male? Are being male and wearing jeans disjoint? Are gender and attire independent?

Copyright © 2010 Pearson Education, Inc. Building a contingency table Drivers suspected of drunk driving are given tests to determine blood alcohol content. 78% of suspects get a breath test, 36% a blood test, and 22% get both. Are any of these numbers marginal probabilities? Are any joint probabilities? Slide Breath test No breath test total Blood test No blood test total

Copyright © 2010 Pearson Education, Inc. Building a contingency table Are giving a blood test and a breath test mutually exclusive? Are giving the two tests independent? Slide Breath test No breath test total Blood test No blood test total

Copyright © 2010 Pearson Education, Inc. In our text book, 48% of pages have a data display, 27% have an equation, 7% have both. Make a contingency table for the variables display and equation. What is the probability for a randomly selected page with an equation to also have a data display? Are these disjoint events? Are these independent events? Slide

Copyright © 2010 Pearson Education, Inc. Slide Tree Diagrams A tree diagram helps us think through conditional probabilities by showing sequences of events as paths that look like branches of a tree. Making a tree diagram for situations with conditional probabilities is consistent with our “make a picture” mantra.

Copyright © 2010 Pearson Education, Inc. Slide Tree Diagrams (cont.) Figure 15.5 is a nice example of a tree diagram and shows how we multiply the probabilities of the branches together. All the final outcomes are disjoint and must add up to one. We can add the final probabilities to find probabilities of compound events.

Copyright © 2010 Pearson Education, Inc. Making a tree diagram A recent study found that in 77% of all accidents the driver was wearing a seatbelt. 92% of those drivers escaped serious injury, while only 63% of those not wearing seatbelts escaped injury. Construct a tree diagram. What is the probability that someone is seriously injured in an accident? What is the probability that a driver who was seriously injured wasn’t wearing a seatbelt? Slide

Copyright © 2010 Pearson Education, Inc. In April 2003, Science magazine reported on a new computer-based test for ovarian cancer, “clinical proteomics,” that examines a blood sample for the presence of certain patterns of proteins. Ovarian cancer, though dangerous, is very rare, afflicting only 1 of every 5000 women. The test is highly sensitive, able to correctly detect the presence of ovarian cancer in 99.97% of women who have the disease. However, it is unlikely to be used as a screening test in the general population because the test gave false positives 5% of the time. Why are false positives such a big problem? Draw a tree diagram and determine the probability that a woman who tests positive using this method actually has ovarian cancer. Slide

Copyright © 2010 Pearson Education, Inc. Slide What Can Go Wrong? Don’t use a simple probability rule where a general rule is appropriate: Don’t assume that two events are independent or disjoint without checking that they are. Don’t find probabilities for samples drawn without replacement as if they had been drawn with replacement. Don’t reverse conditioning naively. Don’t confuse “disjoint” with “independent.”

Copyright © 2010 Pearson Education, Inc. Slide What have we learned? The probability rules from Chapter 14 only work in special cases—when events are disjoint or independent. We now know the General Addition Rule and General Multiplication Rule. We also know about conditional probabilities and that reversing the conditioning can give surprising results.

Copyright © 2010 Pearson Education, Inc. Slide What have we learned? (cont.) Venn diagrams, tables, and tree diagrams help organize our thinking about probabilities. We now know more about independence—a sound understanding of independence will be important throughout the rest of this course.