1. From Randomness to Probability CHAPTER 14

2. Randomness 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.

3. Probability The probability of an event is its long-run relative frequency. ◦While we may not be able to predict a particular individual outcome, we can talk about what will happen in the long run. For any random phenomenon, each attempt, or trial, generates an outcome. ◦Something happens on each trial, and we call whatever happens the outcome. ◦These outcomes are individual possibilities, like the number we see on top when we roll a die.

4. Probability Sometimes we are interested in a combination of outcomes (e.g., a die is rolled and comes up even). A combination of outcomes is called an event. 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

5. 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 the true relative frequency as the number of trials increases. ◦For example, consider flipping a fair coin many, many times. The overall percentage of heads should settle down to about 50% as the number of outcomes increases.

6. LLN The common (mis)understanding of the LLN is that random phenomena are supposed to compensate some for whatever happened in the past. This is just not true. For example, when flipping a fair coin, if heads comes up on each of the first 10 flips, what do you think the chance is that tails will come up on the next flip? ◦For example, when flipping a fair coin, if heads comes up on each of the first 10 flips, what do you think the chance is that tails will come up on the next flip?

7. Probability Thanks to the LLN, we know that relative frequencies settle down in the long run, so we can officially give the name probability to that value. Probabilities must be between 0 and 1, inclusive. ◦A probability of 0 indicates impossibility. ◦A probability of 1 indicates certainty.

8. Equally likely outcomes 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. Personal Probabilities In everyday speech, when we express a degree of uncertainty without basing it on long-run relative frequencies, 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.

10. Formal probability Two Requirements for a probability ◦A probability is a number between 0 and 1. ◦For any event A, 0 ≤ P(A) ≤ 1.

11. Rules The rules of Probability 1. Something has to happen rule A probability of zero means an event never happens A probability of one means an event always happens

12. Formal Probability continued 2.Something has to happen rule”: ◦The probability of the set of all possible outcomes of a trial must be 1. ◦P(S) = 1 (S represents the set of all possible outcomes.)

13. Formal Probability continued 3.Complement Rule:  The set of outcomes that are not in the event A is called the complement of A, denoted A C.  The probability of an event occurring is 1 minus the probability that it doesn’t occur: P(A) = 1 – P(A C )

14. Example 1: List the sample space and tell whether the outcomes are equally likely. A family has two children; record the genders in order of birth. Answer: BB, BG, GB, GG Yes equally likely

15. Example 2: The plastic arrow on a spinner for a child's game stops rotating to point at a color that will determine what happens next. Determine whether the following probability assignment is legitimate. Red Yellow Green Blue 0.50 0.20 0.15 0.15 Answer: 0.50 + 0.20 + 0.15 + 0.15 = 1.00 Yes

16. Example 3 A fair coin has come up "heads" 10 times in a row. The probability that the coin will come up heads on the next flip is... Answer: 0.50 Each coin flip is independent of the other coin flips

17. Example 4 In a survey of American women who were asked to name their favorite color, 19% said blue, 19% said red, 16% said green, 11% said yellow, 14% said black, and the rest named another color. If you pick a survey participant at random, what is the probability that she named another color? Answer: 1 – (0.19 + 0.19 + 0.16 + 0.11 + 0.14) = ◦= 1 -.79 ◦=.21

18. Addition Rule ◦Events that have no outcomes in common (and, thus, cannot occur together) are called disjoint (or mutually exclusive).

19. Addition rule continued ◦For two disjoint events A and B, the probability that one or the other occurs is the sum of the probabilities of the two events. ◦P(A or B) = P(A) + P(B), provided that A and B are disjoint. ◦Also written as P(A  B) instead of P(A or B)

20. Multiplication Rule ◦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. ◦P(A and B) = P(A) x P(B), provided that A and B are independent. ◦ P(A  B) instead of P(A and B)

21. Multiplication rule continued ◦Two independent events A and B are not disjoint, provided the two events have probabilities greater than zero:

22. Example M & M’s In 2001 the company that makes m & m’s surveyed nearly every country in the world to vote on a new color between purple, pink and teal. Purple was the global winner, however in Japan the percentages were 38% pink, 36% teal, and only 16% purple. 1.What’s the probability that a Japanese M & M’s survey respondent selected at random preferred either pink or teal? Answer 0.38 + 0.36 = 0.74 Addition rule

23. Example 2. If we pick two respondents at random, what’s the probability that they both selected purple? Answer: 0.16 * 0.16 = 0.0256 Multiplication rule 3. If we pick three respondents at random, what's the probability that at least one preferred purple? Answer: These are 3 independent events P(not purple) = 1 – P(purple) = 1- 0.16 = 0.84 So P(none picked purple) = (0.84) = 0.5927 P(at least one picked purple) = 1 – P(none picked purple) = 1 – 0.5927 = 0.4073

24. Probability Rules Chapter 15 For any random phenomenon, each trial generates an outcome. An event is any set or collection of outcomes. The collection of all possible outcomes is called the sample space, denoted S.

25. Events When outcomes are equally likely, probabilities for events are easy to find just by counting. When the k possible outcomes are equally likely, each has a probability of 1/k. For any event A that is made up of equally likely outcomes,

26. The rule with dealing with the probability rules Make a picture !!! The most common type of picture to make is a Venn Diagram.

27. The General Addition rule When two events are disjoint (mutually exclusive) we use: However when two events are not disjoint we use the General Addition rule P(A or B ) = P(A) + P(B) – P(A and B)

28. Conditional Probabilities 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.

29. Conditional Probabilities 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.

30. General Multiplication Rule When two events are not independent we use the General Multiplication rule For any two events A and B P(A and B) = P(A) x P(B|A) or P(A and B) = P(B) x P(A|B)

31. 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).)

32. 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.

33. Slide 15- 33 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.

34. Slide 15- 34 Tree Diagrams (cont.) Figure 15.4 is a nice example of a tree diagram and shows how we multiply the probabilities of the branches together:

35. Example A survey reveals that 34% of people are entertained by reading a book, 47% of people are entertained by watching TV and 19% are entertained by books or TV. What is the probability that a person is entertained by either reading a book or watching TV? P(books or TV) = P(Books) + P(TV) – P(Books and TV) =.34 + 47 -.19 =.62

36. Example

37. The table shows the political affiliation of voters in one city and their opinion on strong gun control laws. What is the probability that a democrat opposes stronger gun control laws? FavorOppose Republican0.070.33 Democrat0.220.16 Other0.140.08

38. Example continued

39. Reversing the condition Suppose we want to know P(A|B), but we know only P(A), P(B), and P(B|A). We also know P(A and B), since P(A and B) = P(A) x P(B|A) From this information, we can find P(A|B): This is known as Bayes’s Rule

40. Example 3.3% of a population is infected with a certain disease. There is a test for the disease, however the test is not completely accurate. 94% of those who have the disease will test positive. However, 4.4% of those who do not have the disease will also test positive (a false positive). What is the probability that any given person will test positive? What is the probability that a person tested positive will be infected?

41. 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.”

42. 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.

43. What have we learned, continued 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.