2. Copyright © 2009 Pearson Education, Inc. Chapter 14 From Randomness to Probability

3. Copyright © 2009 Pearson Education, Inc. Slide 1- 3 Objectives: State the definition of sample space, event and P(A). Apply the Law of Large Numbers. Recognize when events are disjoint and when events are independent. State the basic definitions and apply the rules of probability for disjoint and independent events.

4. Copyright © 2009 Pearson Education, Inc. Slide 1- 4 Dealing with Random Phenomena A random phenomenon is a situation in which we know what outcomes could happen, but we don’t know which particular outcome did or will happen. In general, each occasion upon which we observe a random phenomenon is called a trial. At each trial, we note the value of the random phenomenon, and call it an outcome. When we combine outcomes, the resulting combination is an event. The collection of all possible outcomes is called the sample space.

5. Copyright © 2009 Pearson Education, Inc. Slide 1- 5 First a definition... When thinking about what happens with combinations of outcomes, things are simplified if the individual trials are independent. Roughly speaking, this means that the outcome of one trial doesn’t influence or change the outcome of another. For example, coin flips are independent. Approach #1: Empirical Probability (i.e. Relative Frequency Approach)

6. Copyright © 2009 Pearson Education, Inc. Slide 1- 6 The Law of Large Numbers The Law of Large Numbers (LLN) says that the long-run relative frequency of repeated independent events gets closer and closer to a single value. We call the single value the probability of the event. Because this definition is based on repeatedly observing the event’s outcome, this definition of probability is often called empirical probability. Example: If we flip a fair coin repeatedly, over time we expect to get heads ½ of the time.

7. Copyright © 2009 Pearson Education, Inc. Slide 1- 7 The Nonexistent Law of Averages The LLN says nothing about short-run behavior. Relative frequencies even out only in the long run, and this long run is really long (infinitely long, in fact). The so called Law of Averages (that an outcome of a random event that hasn’t occurred in many trials is “due” to occur) doesn’t exist at all. Example: suppose a couple has 3 children all of whom are boys, is the couple more likely to have a girl for the next child? Or suppose a group of pregnant friends find that 5 of the 6 of them are expecting girls, is the 6 th friend more likely to have a boy?

8. Copyright © 2009 Pearson Education, Inc. Slide 1- 8 Approach #2: Classical Approach - Modeling Probability When probability was first studied, a group of French mathematicians looked at games of chance in which all the possible outcomes were equally likely. It’s equally likely to get any one of six outcomes from the roll of a fair die. It’s equally likely to get heads or tails from the toss of a fair coin. However, keep in mind that events are not always equally likely. A skilled basketball player has a better than 50-50 chance of making a free throw.

9. Copyright © 2009 Pearson Education, Inc. Slide 1- 9 Given equally likely outcomes, the probability of an event is the number of outcomes in the event divided by the total number of possible outcomes. P(A) = Dice roll examples: P(rolling a 5) P(rolling a number less than 3) Modeling Probability (cont.) # of outcomes in A # of possible outcomes

10. Copyright © 2009 Pearson Education, Inc. Slide 1- 10 Approach #3: Subjective or Personal Probability In everyday speech, when we express a degree of uncertainty without basing it on long-run relative frequencies or mathematical models, we are stating subjective or personal probabilities. Personal probabilities don’t display the kind of consistency that we will need probabilities to have, so we’ll stick with formally defined probabilities.

11. Copyright © 2009 Pearson Education, Inc. Slide 1- 11 The First Three Rules of Working with Probability We are dealing with probabilities now, not data, but the three rules don’t change. Make a picture.

12. Copyright © 2009 Pearson Education, Inc. Slide 1- 12 The First Three Rules of Working with Probability (cont.) The most common kind of picture to make is called a Venn diagram. We will see Venn diagrams in practice shortly…

13. Copyright © 2009 Pearson Education, Inc. Slide 1- 13 Formal Probability Probability Boundaries: Rule 1) A probability is a number between 0 and 1. For any event A, 0 ≤ P(A) ≤ 1. The probability of an impossible event is 0 The probability of a certain event is 1

14. Copyright © 2009 Pearson Education, Inc. Slide 1- 14 Formal Probability (cont.) Rule 2) The probability of the set of all possible outcomes of a trial must be 1. Otherwise the probabilities are not legitimate P(S) = 1 (S represents the set of all possible outcomes of a trial.)

15. Copyright © 2009 Pearson Education, Inc. Complementary Events Rule 3) The complement of event A, denoted by A C, consists of everything in the sample space S except event A P(A) + P(A C ) = 1 P(A C ) = 1 – P(A) P(A) = 1 – P(A C ) Examples: When flipping a coin, the complement of “heads” is “tails” When rolling a die, the complement of “rolling a 1 or 2” is “rolling a 3, 4, 5, or 6”

16. Copyright © 2009 Pearson Education, Inc. Mutually Exclusive Events Events A and B are mutually exclusive (disjoint) if they cannot occur at the same time (that is, they don’t overlap or have any outcomes in common) If events A and B are mutually exclusive, then: P(A and B) = 0 P(A and B) is the probability that event A occurs and event B occurs

17. Copyright © 2009 Pearson Education, Inc. Addition Rule (“Light” Version) P(A or B) is the probability that event A occurs or event B occurs or both occur If A and B are mutually exclusive events, then: P(A or B) = P(A) + P(B) This rule can be extended to more than two mutually exclusive events

18. Copyright © 2009 Pearson Education, Inc. Slide 1- 18 Formal Probability (cont.) 4.Addition Rule Examples: P(A or B) = P(A) + P(B), provided that A and B are disjoint. Examples: suppose you roll a die. What is the probability that you roll a 5 or 6? Are these events disjoint? What about the probability that you roll a even number or a number less than 3? Are these events disjoint?

19. Copyright © 2009 Pearson Education, Inc. Independent Events Events A and B are independent if the outcome of one event doesn’t influence the outcome of the other

20. Copyright © 2009 Pearson Education, Inc. Multiplication Rule (“Light” Version) For two independent events A and B, the probability that both A and B occur is the product of the probabilities of the two events. That is, P(A and B) = P(A)*P(B) This rule can be extended to more than two independent events

21. Copyright © 2009 Pearson Education, Inc. Slide 1- 21 Formal Probability 5.Multiplication Rule (cont.): P(A and B) = P(A) x P(B), provided that A and B are independent. Example: the outcomes of rolling 2 distinct dice are independent. What is the probability of rolling a 1 then a 2? What is the probability of rolling an even number on the first die and then a number greater than 4 on the second? What is the probability of rolling a 5 four times in a row?

22. Copyright © 2009 Pearson Education, Inc. Slide 1- 22 Formal Probability (cont.) 5.Multiplication Rule: Many Statistics methods require an Independence Assumption, but assuming independence doesn’t make it true. Always Think about whether that assumption is reasonable before using the Multiplication Rule.

23. Copyright © 2009 Pearson Education, Inc. Slide 1- 23 Formal Probability - Notation Notation alert: In this text we use the notation P(A or B) and P(A and B). In other situations, you might see the following: P(A  B) instead of P(A or B) P(A  B) instead of P(A and B)

24. Copyright © 2009 Pearson Education, Inc. Slide 1- 24 Putting the Rules to Work In most situations where we want to find a probability, we’ll often use the rules in combination. A good thing to remember is that sometimes it can be easier to work with the complement of the event we’re really interested in.

25. Copyright © 2009 Pearson Education, Inc. 2) Sample Spaces: list the sample space and tell whether the events are equally likely A) Toss 2 coins; record the order of heads and tails B) A family has 3 children; record the number of boys C) Flip a coin until you get a head or 3 consecutive tails D) Roll two dice; record the larger number Slide 1- 25

26. Copyright © 2009 Pearson Education, Inc. 11) the plastic arrow on a spinner for a child’s game stops rotating to point at a color that determines what will happen next. Which of the following probability assignments are possible? RedYellowGreenBlue a) 0.25 0.250.250.25 b) 0.100.200.300.40 c) 0.200.300.400.50 d) 001.000 e) 0.100.201.20-1.50 Slide 1- 26

27. Copyright © 2009 Pearson Education, Inc. 14) Funding for many schools comes from taxes based on assessed values of local properties. People’s homes are assessed higher if hey have extra features such as garages and swimming pools. Assessment records in a certain school district indicate that 37% of the residential properties have garages and 3% have swimming pools. The Addition Rule might suggest, then, that 40% of residences have a garage or a pool. What’s wrong with this reasoning? Slide 1- 27

28. Copyright © 2009 Pearson Education, Inc. 16) Lefties. Although its hard to be definitive in classifying people as right or left handed, some studies suggest that about 14% of people are left handed. Since 0.14x0.14 = 0.0196, the Multiplication Rule might suggest that there’s about a 2% chance that a brother and sister are both lefties. What’s wrong with that reasoning? Slide 1- 28

29. Copyright © 2009 Pearson Education, Inc. 17 and 18) In 2007 Harvard accepted about 9% of its applicants, Stanford 10% and Penn 16%. Jorge has applied to all three. Assuming that he is a typical applicant, he figures that his chance of getting into both Harvard and Stanford must be about 0.9% How has he arrived at this conclusion? What additional assumption is he making? Do you agree with his conclusion He figures that his chances of getting into at least one of the three must be about 35% How has he arrived at this conclusion? What assumption is he making? Do you agree with his conclusion? Slide 1- 29

30. Copyright © 2009 Pearson Education, Inc. In a large introductory statistics class, the professor reports that 55% of the students have never taken Calculus, 32% have taken one semester of Calculus, and the rest have taken two or more semesters of Calculus. The professor randomly assigns students to work in groups of three. What is the probability that the first groupmate you meet has studied 2+ semesters of Calculus? Some Calculus? No more than one semester of Calculus? What is the probability that, of your two groupmates Neither has studied Calculus? Both have studied at least one semester of Calculus? At least one has had more than one semester of Calculus? Slide 1- 30

31. Copyright © 2009 Pearson Education, Inc. 32) The American Red Cross says that about 45% of the U.S. population has Type O blood, 40% Type A, 11% Type B, and the rest Type AB. Someone volunteers to give blood, what is the probability that this donor Has type AB blood? Has type A or Type B blood? Is not Type O? Among four potential donors, what is the probability that All are type O? No one is Type AB? They are not all Type A? At least one person is Type B? Slide 1- 31

32. Copyright © 2009 Pearson Education, Inc. 34) The American Red Cross says that about 45% of the U.S. population has Type O blood, 40% Type A, 11% Type B, and the rest Type AB. If you examine one person, are the events that the person is Type A and that the person is Type B disjoint, independent, or neither? If you examine two people, are the events that the first is Type A and the second Type B disjoint, independent, or neither? Can disjoint events ever be independent? Slide 1- 32

33. Copyright © 2009 Pearson Education, Inc. Slide 1- 33 What Can Go Wrong? Beware of probabilities that don’t add up to 1. To be a legitimate probability assignment, the sum of the probabilities for all possible outcomes must total 1. Don’t add probabilities of events if they’re not disjoint. Events must be disjoint to use the Addition Rule.

34. Copyright © 2009 Pearson Education, Inc. Slide 1- 34 What Can Go Wrong? (cont.) Don’t multiply probabilities of events if they’re not independent. The multiplication of probabilities of events that are not independent is one of the most common errors people make in dealing with probabilities. Don’t confuse disjoint and independent

35. Copyright © 2009 Pearson Education, Inc. Slide 1- 35 What have we learned? Probability is based on long-run relative frequencies. The Law of Large Numbers speaks only of long- run behavior. Watch out for misinterpreting the LLN.

36. Copyright © 2009 Pearson Education, Inc. Slide 1- 36 What have we learned? (cont.) There are some basic rules for combining probabilities of outcomes to find probabilities of more complex events. We have the: Probability Assignment Rule Complement Rule Addition Rule for disjoint events Multiplication Rule for independent events