Probability

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## Probability

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**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)**Probability Overview**• Random Generating Processes • Probability Properties • Probability Rules • Example: Binomial Random Processes**Types of Explanations**Data could be generated by: • A purely systematic process • A purely random process • A combination of systematic and random processes**Types of Explanations**Data could be generated by: • A purely systematic process • A purely random process • A combination of systematic and random processes**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.**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.**Random Generating Processes**• Random is not the same as haphazard or helter-skelter or higgledy-piggledy. • Random generating processes yield “characteristic properties of uncertainty”.**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.**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**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**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.**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**“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)**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?**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**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?**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**P = .14%**USA Today survey of 966 inventors who hold U.S. patents.**More Probability Properties**Unconditional Probability • The general probability (relative frequency) of an event, in the absence of any other information**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.**.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?**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**.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)**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.**.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)**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)**“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**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**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**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**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**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?**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**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)**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**- 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**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**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**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**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**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**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)**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**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.