Chapter 6

1. Chapter 6 Probability

2. Inferential Statistics • Samples - so far we have been concerned about describing and summarizing samples or subsets of a population • Inferential stats allows us to “go beyond” our sample and make educated guesses about the population

3. Inferential Statistics • But, we need some help which comes from Probability theory Inferential = Descriptive + Probability Statistics Statistics Theory

4. What is probability theory and what is its role? • Probability theory, or better “probability theories” are found in mathematics and are interested in questions about unpredictable events • Although there is no universal agreement about what probability is, probability helps us with the possible outcomes of random sampling from populations

5. Examples • “The Chance of Rain” • The odds of rolling a five at the craps table (or winning at black jack) • The chance that a radioactive mass will emit a particle • The probability that a coin will come up heads upon flippingit* • The probability of getting exactly 2 heads out of 3 flips of a coin* • The probability of getting 2 or more heads out of 3 flips*

6. Set Theory (a brief digression) • Experiment - an act which leads to an unpredictable, but measurable outcome • Set - a collection of outcomes • Event - one possible outcome; a value of a variable being measured • Simple probability – the likelihood that an event occurs in a single random observation

7. Simple Probabilities • To compute a simple probability (read the probability of some event), p(event):

8. Probability Theory • Most of us understand probability in terms of a relative frequency measure (remember this, f/n), the frequency of occurrences (f) divided by the total number of trials or observations (n) • However, probability theories are about the properties of probability not whether they are true or not (the determination of a probability can come from a variety of sources)

9. Example 100 marbles are Placed in a jar

10. Relative Frequency (a reminder) • What if we counted all of the marbles in the jar and constructed a frequency distribution? • We find 50 black marbles, 25 red marbles, and 5 white marbles • Relative frequency (proportion) seems like probability Color f rf Black 50 .50 Red 25 .25 White 25 .25 Total 100 1.00

11. Relative Frequency and Probability Distributions • A graphical representation of a relative frequency distribution is also similar to a “Probability Distribution”

12. What does this mean? • What will happen if we choose a single marble out of the jar? • If we chose 100 marbles from the jar, tallied the color, and replaced them, will we get 50, 25, and 25? If so, what if we selected only 99? • If .5, .25, and .25 are the “real” probabilities, then “in the long run” will should get relative proportions that are close to .5, .25, and .25

13. Bernoulli’s Theorem • The notion of “in the long run”is attributed to Bernoulli • It is also known as the “law of large numbers” • as the number of times an experiment is performed approaches infinity (becomes large), the “true” probability of any outcome equals the relative proportion

14. Venn Diagrams A A S

15. Venn Diagrams A “not red” A “Red” S “all the marbles”

16. Mutually Exclusive Events A B S

17. Mutually Exclusive Events

18. Axiom’s of Probability 1. The probability of any event A, denoted p(A), is 0 < p(A) < 1 2. The probability of S, or of an event in sample space S is 1 3. If there is a sequence of mutually exclusive events (B1, B2, B3, etc.) and C represents the event “at least one of the Bi’s occurs, then the probability of C is the sum of the probabilities of the Bis (p(C) = Σp(Bi)

19. 1. 0 < p(A) < 1 (in Venn diagrams) A S S A The probability of event A is between 0 and 1

20. 2. p(S) = 1 A The probability of AN event, in S, occurring is 1 S

21. 3. p(C) = Σp(Bi) If the events B1, B2, B3, etc. are mutually exclusive, the probability of one of the Bs occurring is C, the sum of the Bs B1 B5 C B4 B2 B3 S

22. Mutually Exclusive Events • If A and B are mutually exclusive, meaning that an event of type A precludes event B from occurring, by the 3rd axiom of probability

23. Mutually Exclusive Events • If A and B are mutually exclusive, and set A and set B are not null sets,

24. Joint Events • If the events are independent, (not mutually exclusive), meaning that the occurrence of one does not affect the occurrence of the other, the intersection

25. Joint Events - Example • What is the probability of selecting a black marble and white marble in two successive selections? • Since each selection is independent, then p(Black, White) = .5 • .25 = .125

26. Generalization from Joint Events • If A, B, C, and D are independent events, then: What is the probability of selecting a white marble, then red, then white, then black?

27. What if the events are not independent? • Conditional probability - the occurrence of one event is influenced by another event • “Conditional Probability” refers to the probability of one event under the condition that the other event is known to have occurred • p(A | B) - read “the probability of A given that B has occurred”

28. Probability Theory and Hypothesis Testing • A man comes up to you on the street and says that he has a “special” quarter that, when flipped, comes up heads more often than tails • He says you can buy it from him for $1 • You say that you want to test the coin before you buy it • He says “OK”, but you can only flip it 5 times

29. Probability Theory Example • How many heads would convince you that it was a “special” coin? • 3?, 4?, 5? • How “sure” do you want to be that it is a “special” coin? What is the chance that he is fooling you and selling a “regular” quarter?

30. 2 Hypotheses • The coin is not biased, it’s a normal quarter that you can get at any bank • The likelihood of getting a heads on a single flip is 1/2, or .5 • The coin is a special • The probability of getting a heads on a single flip is greater than .5

31. Hypothesis Testing • Let’s assume that it is a regular, old quarter p = .5 (the probability of getting a heads on a SINGLE toss is .5) • We flip the coin and get 4 heads. What is the probability of this result, assuming the coin is fair? • Note that this is a problem involving conditional probability :p(4/5 heads|coin is fair)

32. How do I solve this problem? • Any Ideas? • You might think that, using the rule of Joint events, that: • NO!

33. Why not? • You have just calculated the probability of getting exactly H, H, H, H on four flips of our coin. • What is the probability of getting H, T, H, T on 4 flips? • Exactly the same as H, H, H, H…any single combination of 4 H and T are equally likely in this scenario. • Here they are:

34. All Possibilities: 4 flips of a coin 0 Heads 1 heads 2 heads 3 heads 4 heads f = 1 4 6 4 1

35. Relative Frequency Dist

36. YES! • Relative frequency and Probability are related by Bernoulli’s theorem • If I did this test again, would I get the same result? (probably not) • If I did it over and over again, what results would we expect given a non-biased coin? • How many combinations?

37. What if • I figured out the total number of possible outcomes of this experiment, • and I figured out the total number of outcomes that had 4/5 heads? • Prob of 4/5 = Freq of 4/5 Total N of outcomes How many outcomes?

38. Lots • H, H, T, T, T • T, H, T, H, T • H, T, H, T, H • ETC. ETC. ETC.

39. How many 4 out of 5? • 5 flips • (exactly) 4 heads • 1 possibility – H, H, H, H, T • Another – H, H, H, T, H • More – H, H, T, H, H • And – H, T, H, H, H • Lastly – T, H, H, H, H

40. 5 Flips: All possibilities 0 heads 1 head 2 heads 3 heads 4 heads 5 heads p = .03125 .15625 .3125 .3125 .15625 .03125

41. At least 4 heads out of 5 • Given a Fair Coin: • Getting at least 4 heads out of 5 flips is p(4) + p(5) .15625+.03125 = .1875 There is a 18.75% chance that, upon flipping a FAIR coin 5 times, you will get at least 4 heads. B1 B5 C B4 B2 B3

42. You gonna buy that quarter? • What if this guy let you flip this quarter 100 times? • How many times do you want to flip it? • (the more the better, yes? In the long run???)