Announcements. Lecture 5: Probability. Dangling threads from last week: Mean vs. median. Dangling threads from last week: Sampling bias

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Recap Announcements Lecture 5: Statistics 101 Mine Çetinkaya-Rundel September 13, 2011 HW1 due TA hours Thursday - Sunday 4pm - 9pm at Old Chem 211A If you added the class last week please make sure

1 Recap Announcements Lecture 5: Statistics 101 Mine Çetinkaya-Rundel September 13, 2011 HW1 due TA hours Thursday - Sunday 4pm - 9pm at Old Chem 211A If you added the class last week please make sure to review the syllabus Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27 Recap Dangling threads from last week: Mean vs. median Scores on an exam: 79, 82, 83, 92, 94 x = 86, median = 83 Scores on an exam w/ extreme observation: 3, 79, 82, 83, 92, 94 x = 72.2, median = 82.5 When an extreme observation is included in the data set, the typical value can be better estimated by the median. Recap Dangling threads from last week: Sampling bias Non-response bias occurs when some of the randomly sampled individuals are unwilling or unable to respond and those who respond differ in meaningful ways from those who do not. Convenience sample bias occurs when the randomly sampled members are only from a convenient portion of the population. This is a common problem with businesses only asking their customers opinions when their target population includes those who are not already their customers. Voluntary response bias occurs when responses come from self-selected volunteers. Such polls are usually not considered to be scientific. Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27 Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27

2 Recap Review of commonly missed questions on the online quiz Recap (graded) Suppose we want to estimate family size, where family is defined as married couples and their children, if any. If we select students at random at an elementary school and ask them what their family size is, will our average be biased? If so, will it overestimate or underestimate the true value? (a) not biased (b) biased, underestimate (c) biased, overestimate Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27 Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27 Rolling a die - success: Rolling a 1 There are several possible interpretations of probability but they (almost) completely agree on the mathematical rules probability must follow. P(A) = of event A 0 P(A) 1 Frequentist interpretation: The probability of an outcome is the proportion of times the outcome would occur if we observed the random process an infinite number of times. Single main stream school until recently. Bayesian interpretation: A Bayesian interprets probability as a subjective degree of belief: For the same event, two separate people could have differing probabilities. Largely popularized by revolutionary advance in computational technology and methods during the last twenty years Cumulative frequency of rolls that are n (number of rolls) Roll Success Cumulative Frequency Cumulative Proportion 0.3 Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27 Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27

Law of large numbers Law of large numbers states that as more observations are collected, the proportion of occurrences with a particular outcome, ˆp n, converges to the probability of that outcome,

3 Law of large numbers Law of large numbers states that as more observations are collected, the proportion of occurrences with a particular outcome, ˆp n, converges to the probability of that outcome, p Which of the following outcomes would you be most surprised by? p^n (a) 3 heads in 10 coin flips (b) 3 heads in 100 coin flips (c) 3 heads in 1000 coin flips ,000 10, ,000 n (number of rolls) Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27 Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27 Law of large numbers (cont.) When tossing a fair coin, if heads comes up on each of the first 10 tosss, what do you think the chance is that another head will come up on the next toss? H H H H H H H H H H? The probability is still 0.5, or there is still a 50% chance that another head will come up on the next toss. P(H on 11 th toss) = P(T on 11 th toss) = 0.5 The coin is not due for a tail. The common (mis)understanding of the LLN is that random processes are supposed to compensate for whatever happened in the past; this is just not true and is also called gambler s fallacy (or law of averages). Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27 Random processes A random process is a situation in which we know what outcomes could happen, but we don t know which particular outcome will happen. Examples: coin tosses, die rolls, itunes shuffle, whether the stock market goes up or down tomorrow, etc. It can be helpful to model a process as random even if it is not truly random. Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27

4 Disjoint/mutually exclusive Disjoint or mutually exclusive outcomes Addition rule of disjoint outcomes Disjoint or mutually exclusive outcomes Two outcomes are called disjoint or mutually exclusive if they cannot both happen. The outcome of a coin toss cannot be a head and a tail. A student cannot fail and pass a class. A card drawn from a deck cannot be an ace and a queen. When events are disjoint, it s easy to calculate the probability of one event or the other happening. The probability of rolling a 1 or a 2: Addition rule of disjoint outcomes P(A or B) = P(A) + P(B) Or more generally, P(A 1 or or A k ) = P(A 1 ) + + P(A k ) P(1) + P(2) = = The probability of tossing a head or a tail: P(H) + P(T ) = = 1 Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27 Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27 Probabilities when events are not disjoint Probabilities when events are not disjoint Non-disjoint events What is the probability of drawing a jack or a red card from a well shuffled full deck? P(jack or red) = P(jack) + P(red) P(jack and red) = = What is the probability that a randomly sampled Sta 101 student has a job at Duke or has an excellent experience here? Job Yes No Total Excellent Experience Satisfactory Unsatisfactory Total General addition rule P(A or B) = P(A) + P(B) P(A and B) (a) 29/126 (b) 97/126 (c) 49/126 (d) 29/49 Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27 Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27

5 distributions distributions distributions A probability distribution lists the possible outcomes and the probabilities with which they occur. The probability distribution for outcomes of a coin toss: Event Head 0.5 Tail 0.5 Rules for probability distributions: 1 The outcomes listed must be disjoint. 2 Each probability must be between 0 and 1. 3 The probabilities must total 1. In a survey, 52% of respondents said they are Democrats. What is the probability that a randomly selected respondent from this sample is a Republican? (a) 0.48 (b) more than 0.48 (c) less than 0.48 (d) cannot calculate from the information given Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27 Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27 Complement of an event Complement of an event Sample space Complementary events Sample space is the collection of all possible outcomes of a trial. A couple has one kid, what is the sample space for the gender of this kid? S = { B, G } The complement of event A, A C, represents all outcomes not in A. A couple has one kid. If we know that the kid is not a boy, what is gender of this kid? { B, G } A couple has two kids, if we know that they are not both girls, what are the possible gender combinations for these kids? A couple has two kids, what is the sample space for the gender of these kids? Complementary events P(A) = 1 P(A C ) Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27 Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27

6 Complement of an event Disjoint vs. complementary Do the sum of probabilities of two disjoint events always add up to 1? Do the sum of probabilities of two complementary events always add up to 1? Two processes are independent if knowing the outcome of one provides no useful information about the outcome of the other. Knowing that the coin landed on a head on the first toss does not provide any useful information for determining what the coin will land on in the second toss since coin tosses are independent. Knowing that the first card drawn from a deck is an ace does provide useful information for determining the probability of drawing an ace in the second draw. Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27 Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27 (cont.) Product rule for independent processes You toss a coin twice, what is the sample space for the outcomes of these tosses? S = {HH, TT, HT, TH} You toss a coin twice, what is the probability of getting two tails in a row? Since one out of four possible outcomes match this definition, the probability is 1 4. While it is possible to calculate this probability by first obtaining the sample space, this would be an inefficient way if the number of trials was much higher. Product rule for independent processes P(A and B) = P(A) P(B) Or more generally, P(A 1 and and A k ) = P(A 1 ) P(A k ) Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27 Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27

7 At least 1 According to an August 2011 Gallup poll, 32% of Americans would prefer to work for a male boss and 22% would prefer a female boss if they were taking a new job. If we were to randomly select two job applicants, what is the probability that they would both prefer to work for a male boss? You may assume that the pool of applicants for this job is representative of the American population as a whole. If we were to randomly select 5 job applicants, what is the probability that at least one prefers to work with a female boss? If we were to randomly select 5 job applicants, the sample space for the number of applicants who would prefer a female boss would be: S = {0, 1, 2, 3, 4, 5} We are interested in instances where at least one applicant prefers a female boss: S = {0, 1, 2, 3, 4, 5} So we can divide up the sample space intro two categories: S = {0, at least one} Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27 Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27 At least 1 (cont.) Since the probability of the sample space must add up to 1: Prob(at least 1 prefer F ) = 1 Prob(none prefer F ) = 1 [(1 0.22) 5 ] = = At least 1 Roughly 14% of Duke s first-year class is from North Carolina. If we were to randomly sample 3 Duke first-year students, what is the probability that at least one of them will be from North Carolina? (a) almost 0 (b) (c) (d) P(at least one) = 1 P(none) Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27 Statistics 101 (Mine Çetinkaya-Rundel) Lecture 5: September 13, / 27

Random processes. Lecture 17: Probability, Part 1. Probability. Law of large numbers

Random processes. Lecture 17: Probability, Part 1. Probability. Law of large numbers Random processes Lecture 17: Probability, Part 1 Statistics 10 Colin Rundel March 26, 2012 A random process is a situation in which we know what outcomes could happen, but we don t know which particular