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Probability
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1. Probability Some of the Problems about Chance having a great appearance of Simplicity, the Mind is easily drawn into a belief, that their Solution may be attained by the meer Strength of natural good Sense; which generally proving otherwise and the Mistakes occasioned thereby being not unfrequent, ‘tis presumed that a Book of this Kind, which teaches to distinguish Truth from what seems so nearly to resemble it, will be looked upon as a help to good Reasoning - Abraham de Moivre (1667-1754)

2. Probability Overview • Random Generating Processes • Probability Properties • Probability Rules • Example: Binomial Random Processes

3. Types of Explanations Data could be generated by: • A purely systematic process • A purely random process • A combination of systematic and random processes

4. Types of Explanations Data could be generated by: • A purely systematic process • A purely random process • A combination of systematic and random processes

5. Hypothesis testing • We would like to know which of the three explanations is most likely correct • The “purely systematic” explanation is easy to confirm or reject based on a quick look at the data. (rarely fits social science data) • So we’re left trying to assess the question “could a purely random process fully account for this data?” • If not, we’ll accept the more complex (systematic + random) model.

6. Random Generating Processes • To answer that question, we need to understand random generating processes. (The domain of probability mathematics). • Note: most people intuitively over-estimate the role of systematic factors. One reason is that people often have a poor grasp of how random generating processes actually work.

7. Random Generating Processes • Random is not the same as haphazard or helter-skelter or higgledy-piggledy. • Random generating processes yield “characteristic properties of uncertainty”.

8. Random Generating Processes Example: the Binomial random process • We have two possible outcomes (e.g. heads or tails) associated with a specific probability (e.g. 0.50) • We can’t predict with certainty the particular outcome for any trial, but we can describe the per-trial likelihood. • We can’t say too much about the relative frequency of outcomes in the short-run, but we can say a lot about the relative frequencies in the long-run.

9. Random Generating Processes • When we say something can be described by a random generating process, we do not necessarily mean that it is caused by a mystical thing called “chance” • There may be many independent (but unmeasured) systematic factors that combine together to create the observed random probability distribution. E.g. coin tosses • When we say “random” we just mean that we can’t do any better than some basic (but characteristic) probability statements about how the outcomes will vary

10. PROBABILITY • Probabilities are numbers which describe the likelihoods of random events. • The probability of an event corresponds to the per-trial likelihood of that event, as well as the long-run relative frequency of that event. • P(A) means “the probability of event A.” • If A is certain, then P(A) = 1 • If A is impossible, then P(A) = 0

11. CHANCES and ODDS • Chances are probabilities expressed as percents. Chances range from 0% to 100%. • For example, a probability of .75 is the same as a 75% chance. • The odds for an event is the probability that the event happens, divided by the probability that the event doesn’t happen. Odds can be any positive number. • For example, a probability of .75 is the same as 3-to-1 odds.

12. Sample Space • A sample space is a list of all possible outcomes of a random process. • When I roll a die, the sample space is {1, 2, 3, 4, 5, 6}. • When I toss a coin, the sample space is {head, tail}. • An event is one or more members of the sample space. • For example, “head” is a possible event when I toss a coin. Or “number less than four” is a possible event when I roll a die. • Events are associated with probabilities

13. “complement” Probability Properties • All probabilities are between zero and one: • 0 < P(A) < 1 • Something has to happen: • P(Sample space) = 1 • The probability that something happens is one minus the probability that it doesn’t: • P(A) = 1 - P(not A)

14. If equally likely outcomes P (event A) = HHHHHTHTHHTTTHHTHTTTHTTT Total outcomes: Analytic Approach: Theoretical probabilities What is the probability of getting exactly two heads in three coin tosses?

15. A box contains red and blue marbles. One marble is drawn at random from the box. If it is red, you win \$1. If it is blue you lose \$1. You can choose between two boxes. • Box A contains 3 red marbles and 2 blue ones • Box B contains 51 red marbles and 34 blue ones

16. P (event A) = Some Typical Probability Problems • Anja has to pick a four digit pin number. Each digit will be between 0 and 9. What is the probability that she picks a pin number that has exactly one 3 in it? • A certain senior class has 6 students. Two will receive \$500 scholarships. What is the probability that Kim and AJ are the winning pair?

17. If large sample P (event A) = long term relative frequency = • From a random sample of n = 250, 70 students were classified as narcissists. Relative frequency = Relative Frequency Approach: Observed %s What is the probability that a Columbia MBA student is a narcissist? * Justification: The law of large numbers

18. P = .14% USA Today survey of 966 inventors who hold U.S. patents.

19. More Probability Properties Unconditional Probability • The general probability (relative frequency) of an event, in the absence of any other information

20. Conditional Probability • The conditional probability of B, given A, is written as P(B|A). It is the probability of event B, given that A has occurred. • For example, P(short-sleeved shirt| shorts) is the probability that I will put on a short-sleeved shirt, given that I have already decided to wear shorts. • Note that P(B|A) is not the same as P(A|B). • It is very likely that I will wear a short sleeved shirt if I’m going with shorts. It is not necessarily likely that I will wear shorts just because I’m wearing a short sleeved shirt.

21. .59 .47 .72 Sales Approach Survey No Sale Sale 580 Aggressive 580 Passive 686 474 1160 What is the unconditional probability of making a sale? What is the probability of making a sale, given an aggressive approach? What is the probability of making a sale, given a passive approach?

22. Practical Application of Conditional Probability Sensitivity: probability a test is positive, given disease is present False Positive rate: probability a test is positive, given disease is absent False Negative rate: probability a test is negative, given disease is present

23. .85 .15 .28 Medical Test Survey Disease Absent Disease Present 130 Test Result + 70 Test Result - 130 70 200 What is the sensitivity of the test? P(+, given condition present) What is the false negative rate? P(-, given condition present) What is the false positive rate? P(+, given condition absent)

24. Independence • Events A and B are independent if the probability of event B is the same whether or not A has occurred. • If (and only if) A and B are independent, then P(B | A) = P(B | not A) = P(B) • For example, if I am tossing two coins, the probability that the second coin lands heads is always .50, whether or not the first coin lands heads.

25. .24 .24 .24 Superstition Survey Happy Ending No Happy Ending 600 Throw Rice 800 Not Throw Rice 336 1074 1400 Is rice-throwing statistically independent from happy endings? P(happy│throw rice) =? P(happy│no throw) =? P(happy)

26. Conditional Probability • The probability of A, given B • May be larger, smaller, or equal to the unconditional P(A) Joint Probability • The probability that A and B both occur • Use the multiplication rule • Will always be ≤ to the unconditional P(A)

27. “Linda is thirty-one years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in antinuclear demonstrations.” Which is more likely? • Linda is a bank teller • Linda is a bank teller and is active in the feminist movement

28. Probability Rules • Probability of A or B: Addition Rule P(A or B) = P(A) + P(B), when A and B are mutually exclusive • Probability of A and B: Multiplication Rule P(A and B) = P(A) x P(B), when A and B are independent

29. The Addition Rule B A “mutually exclusive” = A and B cannot both happen P (A or B) = P(A) + P(B) Patricia is getting paired up with a big sister from the neighboring high school. If there are 30 student volunteers (9 seniors, 6 juniors, 7 sophomores, and 8 freshmen), what is the probability her big sister is an upperclassman? P(senior or junior) = P(senior) + P(junior) = .30 + .20 = .50

30. The Multiplication Rule B A “independent” = A does not effect the likelihood of B and vice versa P (A and B) = P(A) X P(B) The probability that Am Ex will offer Frank a job is 50%. The probability Citibank will offer him a job is 30%. Am Ex and Citibank are not in contact. What is the likelihood he gets offered both jobs? What is the likelihood he is offered neither job? P(AmEx and Citibank) = P(AmEx) x P(Citibank) = .50 x .30 = .15 P(Not AmEx and Not Citibank) = P(Not AmEx) x P(Not Citibank) = .50 x .70 = .35

31. When A and B are mutually exclusive, this is zero General Addition Rule B For all cases A P (A or B) = P(A) + P(B) – P(A and B) There are 20 people sitting in a café. 10 like tea, 10 like coffee, and 2 people like both tea and coffee. What is the probability that a random person in the café will like tea or coffee? P(tea or coffee) = P(tea)+P(coffee)-P(tea and coffee) = .50+.50-.10 = .90

32. When A and B are independent, this is same as P(B) P(blue1 and blue2)= P(blue1) x P(blue2│blue1) = General Multiplication Rule B For all cases A P (A and B) = P(A) X P(B│A) There are 10 green and 10 blue marbles in a jar. What is the probability that Sue draws two blue marbles in a row?

33. Summary • Addition Rule P(A or B) = P(A) + P(B), when A and B are mutually exclusive P(A or B) = P(A) + P(B) – P(A and B), generally • Multiplication Rule P(A and B) = P(A) x P(B), when A and B are independent P(A and B) = P(A) x P(B│A), generally

34. Example: Binomial Random Processes • Two possible outcomes • Heads or tails • Make basket or miss basket • Fatality, no fatality • With probability p (or 1-p) • Events are independent • Per trial probability is p (or 1-p) • Long run relative frequency is p (or 1-p)

35. H T H H T H T T H T H T T T H T H H H T H H T H T T T T T H T H T H H Example: Binomial Random Processes • Short run relative frequency is NOT necessarily p • Chance is LUMPY

36. - A string of wins “must” mean a hot table Example: Binomial Random Processes • People are bad random number generators, we put in too few “lumps” for our samples • Conversely, people are too quick to draw conclusions of systematicity from observed “lumps” in a sequence

37. Example: Binomial Random Processes • “Representativeness Error” - People expect a small sample to be too representative of of the population or the long run frequency • “Law of Small Numbers” Error - People are overly confident of observed data patterns based on small samples

38. H H H H H H H H H equally likely H H H H H H H H T Example: Binomial Random Processes • More representativeness errors: “The Gambler’s Fallacy” H H H H H H H H ? • People expect tails to be “temporarily advantaged” after a run of heads • But events are independent

39. Which lotto ticket would you buy? equally likely (or unlikely) to win 26 45 8 72 91 26 26 26 26 26 Less likely to be bought Example: Binomial Random Processes • Predicting a specific versus a general pattern • Each specific ticket is equally (un)likely to win • A ticket that “looks like” ticket A (with alternating values) is more likely than one that “looks like” ticket B (with identical values). • But buyer beware! You are betting on a specific ticket, not a general class of tickets

40. What is the probability of getting heads on the second trial and the tails on all other trials? P(T,H) = 0.25 P(T, H, T) = 0.125 P(T, H, T, T) = 0.0625 Example: Binomial Random Processes • Probabilities for specific patterns get smaller as you run more trials

41. What is the probability of getting at least one heads when you toss a coin multiple times? Two tosses: P(HT or TH or HH) = 0.75 Three tosses: P(HTT or THT or TTH or THH or HHT or HHH) = 0.875 Four tosses: 0.9375 Example: Binomial Random Processes • Probabilities for general patterns get larger as you run more trials

42. Example: Binomial Random Processes • Probabilities for general patterns get larger as you run more trials • Compare: • P(at least one accident) when you ride in a car 2x a week • P(at least one accident) when you ride in a car 7x a week • They say P (fatality in airplane crash) < P(fatality in car crash) • But people spend more time in cars • P(airplane fatality in one minute) = P(car fatality in one minute)

43. The “Hot Hand” • The “hot hand” is a belief about conditional probability. People believe shots are not independent. • Gilovich argues that the pattern of data, however, can be well described by a binomial random process • His Evidence: • Independent shots • Two outcomes: basket or missed basket • Player has general probability p of getting a basket • P(basket |miss) = P(basket|basket) = P(basket) • frequence of 4, 5, 6 basket “streaks” no more likely than a binomial process would predict

44. The “Hot Hand” • Are people just deluded? • There are biases in information processing which contribute to the misperception • But also: • P(streak) is greater when p is greater. • Thus, by a binomial process, good players will have more streaks • P(streak) is greater when more shots are taken • players are not more likely to make the next shot if they made the previous shot, but… • turns out players are more likely to take the next shot if they made the previous shot.