Unit 2: Probability and distributions Lecture 1: Probability and conditional probability
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1 Unit 2: Probability and distributions Lecture 1: Probability and conditional probability Statistics 101 Thomas Leininger May 21, 2013
2 Announcements 1 Announcements 2 Probability Randomness Defining probability Law of large numbers Disjoint and non-disjoint outcomes Probability distributions Independence Recap 3 Marginal, joint, conditional Statistics 101
3 Announcements Announcements PS #1 due today PS #2 assigned (due Friday) Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
4 Announcements Visualization of the day blogs/ the-fix/ wp/ 2013/ 01/ 22/ why-republicans-should-stop-talking-about-roe-v-wade/ Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
5 Announcements Visualization of the day wrong, overturn wrong, don't overturn not wrong, overturn not wrong, don't overturn NA blogs/ the-fix/ wp/ 2013/ 01/ 22/ why-republicans-should-stop-talking-about-roe-v-wade/ Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
6 1 Announcements 2 Probability Randomness Defining probability Law of large numbers Disjoint and non-disjoint outcomes Probability distributions Independence Recap 3 Marginal, joint, conditional Statistics 101
7 Randomness 1 Announcements 2 Probability Randomness Defining probability Law of large numbers Disjoint and non-disjoint outcomes Probability distributions Independence Recap 3 Marginal, joint, conditional Statistics 101
8 Randomness 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. itunes-just-how-random-is-random htm Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
9 Defining probability 1 Announcements 2 Probability Randomness Defining probability Law of large numbers Disjoint and non-disjoint outcomes Probability distributions Independence Recap 3 Marginal, joint, conditional Statistics 101
10 Defining probability Probability There are several possible interpretations of probability but they (almost) completely agree on the mathematical rules probability must follow. P(A) = Probability of event A 0 P(A) 1 Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
11 Defining probability Probability There are several possible interpretations of probability but they (almost) completely agree on the mathematical rules probability must follow. P(A) = Probability 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. Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
12 Defining probability Probability There are several possible interpretations of probability but they (almost) completely agree on the mathematical rules probability must follow. P(A) = Probability 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. Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
13 Law of large numbers 1 Announcements 2 Probability Randomness Defining probability Law of large numbers Disjoint and non-disjoint outcomes Probability distributions Independence Recap 3 Marginal, joint, conditional Statistics 101
14 Law of large numbers Question Which of the following events would you be most surprised by? (a) 3 heads in 10 coin flips (b) 3 heads in 100 coin flips (c) 3 heads in 1000 coin flips Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
15 Law of large numbers Question Which of the following events would you be most surprised by? (a) 3 heads in 10 coin flips (b) 3 heads in 100 coin flips (c) 3 heads in 1000 coin flips Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
16 Law of large numbers 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. Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
17 Law of large numbers Law of large numbers (cont.) When tossing a fair coin, if heads comes up on each of the first 10 tosses, what do you think the chance is that another head will come up on the next toss? 0.5, less than 0.5, or more than 0.5? H H H H H H H H H H? Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
18 Law of large numbers Law of large numbers (cont.) When tossing a fair coin, if heads comes up on each of the first 10 tosses, what do you think the chance is that another head will come up on the next toss? 0.5, less than 0.5, or more than 0.5? 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 Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
19 Law of large numbers Law of large numbers (cont.) When tossing a fair coin, if heads comes up on each of the first 10 tosses, what do you think the chance is that another head will come up on the next toss? 0.5, less than 0.5, or more than 0.5? 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. Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
20 Law of large numbers Law of large numbers (cont.) When tossing a fair coin, if heads comes up on each of the first 10 tosses, what do you think the chance is that another head will come up on the next toss? 0.5, less than 0.5, or more than 0.5? 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 (Thomas Leininger) U2 - L1: Probability May 21, / 30
21 Disjoint and non-disjoint outcomes 1 Announcements 2 Probability Randomness Defining probability Law of large numbers Disjoint and non-disjoint outcomes Probability distributions Independence Recap 3 Marginal, joint, conditional Statistics 101
22 Disjoint and non-disjoint outcomes Disjoint and non-disjoint outcomes Disjoint (mutually exclusive) outcomes: Cannot happen at the same time. The outcome of a single 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. Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
23 Disjoint and non-disjoint outcomes Disjoint and non-disjoint outcomes Disjoint (mutually exclusive) outcomes: Cannot happen at the same time. The outcome of a single 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. Non-disjoint outcomes: Can happen at the same time. A student can get an A in Stats and A in Econ in the same semester. Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
24 Disjoint and non-disjoint outcomes Union of non-disjoint events What is the probability of drawing a jack or a red card from a well shuffled full deck? Figure from math/ discrete/ counting/ cardfreq.htm. Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
25 Disjoint and non-disjoint outcomes Union of 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) = = Figure from math/ discrete/ counting/ cardfreq.htm. Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
26 Disjoint and non-disjoint outcomes Recap General addition rule P(A or B) = P(A) + P(B) P(A and B) Note: For disjoint events P(A and B) = 0, hence the above formula simplifies to P(A or B) = P(A) + P(B). Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
27 Disjoint and non-disjoint outcomes Question What is the probability that a randomly sampled STA 101 student thinks marijuana should be legalized or they agree with their parents political views? Parent Politics Legalize MJ No Yes Total No Yes Total (a) (b) (c) (d) (e) * Data from a previous semester. Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
28 Disjoint and non-disjoint outcomes Question What is the probability that a randomly sampled STA 101 student thinks marijuana should be legalized or they agree with their parents political views? Parent Politics Legalize MJ No Yes Total No Yes Total (a) (b) (c) (d) (e) * Data from a previous semester. Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
29 Probability distributions 1 Announcements 2 Probability Randomness Defining probability Law of large numbers Disjoint and non-disjoint outcomes Probability distributions Independence Recap 3 Marginal, joint, conditional Statistics 101
30 Probability distributions Probability distributions A probability distribution lists all possible events and the probabilities with which they occur. The probability distribution for the gender of one kid: Event B G Probability Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
31 Probability distributions Probability distributions A probability distribution lists all possible events and the probabilities with which they occur. The probability distribution for the gender of one kid: Event B G Probability Rules for probability distributions: 1 The events listed must be disjoint 2 Each probability must be between 0 and 1 3 The probabilities must total 1 Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
32 Probability distributions Probability distributions A probability distribution lists all possible events and the probabilities with which they occur. The probability distribution for the gender of one kid: Event B G Probability Rules for probability distributions: 1 The events listed must be disjoint 2 Each probability must be between 0 and 1 3 The probabilities must total 1 The probability distribution for the genders of two kids: Event BB GG BG GB Probability Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
33 Probability distributions Question 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 using only the information given Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
34 Probability distributions Question 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 using only the information given While (a) and (c) are also possible, (b) is definitely not possible Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
35 Independence 1 Announcements 2 Probability Randomness Defining probability Law of large numbers Disjoint and non-disjoint outcomes Probability distributions Independence Recap 3 Marginal, joint, conditional Statistics 101
36 Independence Independence Two processes are independent if knowing the outcome of one provides no useful information about the outcome of the other. Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
37 Independence Independence 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. Outcomes of two tosses of a coin are independent. Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
38 Independence Independence 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. Outcomes of two tosses of a coin 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. Outcomes of two draws from a deck of cards (without replacement) are dependent. Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
39 Independence Question Between January 9-12, 2013, SurveyUSA interviewed a random sample of 500 NC residents asking them whether they think widespread gun ownership protects law abiding citizens from crime, or makes society more dangerous. 58% of all respondents said it protects citizens. 67% of White respondents, 28% of Black respondents, and 64% of Hispanic respondents shared this view. Which of the below is true? Opinion on gun ownership and race ethnicity are most likely (a) complementary (b) mutually exclusive (c) independent (d) dependent (e) disjoint client/ PollReport.aspx?g=a5f460ef-bba9-484b ea26421b Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
40 Independence Question Between January 9-12, 2013, SurveyUSA interviewed a random sample of 500 NC residents asking them whether they think widespread gun ownership protects law abiding citizens from crime, or makes society more dangerous. 58% of all respondents said it protects citizens. 67% of White respondents, 28% of Black respondents, and 64% of Hispanic respondents shared this view. Which of the below is true? Opinion on gun ownership and race ethnicity are most likely (a) complementary (b) mutually exclusive (c) independent (d) dependent (e) disjoint client/ PollReport.aspx?g=a5f460ef-bba9-484b ea26421b Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
41 Independence Checking for independence If P(A B) = P(A), then A and B are independent. Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
42 Independence Checking for independence If P(A B) = P(A), then A and B are independent. P(protects citizens) = 0.58 Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
43 Independence Checking for independence If P(A B) = P(A), then A and B are independent. P(protects citizens) = 0.58 P(protects citizens White) = 0.67 P(protects citizens Black) = 0.28 P(protects citizens Hispanic) = 0.64 Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
44 Independence Checking for independence If P(A B) = P(A), then A and B are independent. P(protects citizens) = 0.58 P(protects citizens White) = 0.67 P(protects citizens Black) = 0.28 P(protects citizens Hispanic) = 0.64 P(protects citizens) varies by race/ethnicity, therefore opinion on gun ownership and race ethnicity are most likely dependent. Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
45 Independence Determining dependence based on sample data If conditional probabilities calculated based on sample data suggest dependence between two variables, the next step is to conduct a hypothesis test to determine if the observed difference between the probabilities is likely or unlikely to have happened by chance. If the observed difference between the conditional probabilities is large, then the hypothesis test will likely be significant. If the observed difference between the conditional probabilities is small, and the sample is large as well, the hypothesis test may be significant. If the sample is small, then it likely will not be. Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
46 Independence Determining dependence based on sample data If conditional probabilities calculated based on sample data suggest dependence between two variables, the next step is to conduct a hypothesis test to determine if the observed difference between the probabilities is likely or unlikely to have happened by chance. If the observed difference between the conditional probabilities is large, then the hypothesis test will likely be significant. If the observed difference between the conditional probabilities is small, and the sample is large as well, the hypothesis test may be significant. If the sample is small, then it likely will not be. We have seen that P(protects citizens White) = 0.67 and P(protects citizens Hispanic) = Under which condition would you be more convinced of a real difference between the proportions of Whites and Hispanics who think gun widespread gun ownership protects citizens? n = 500 or n = 50, 000 Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
47 Independence Product rule for independent events 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 (Thomas Leininger) U2 - L1: Probability May 21, / 30
48 Independence Product rule for independent events P(A and B) = P(A) P(B) Or more generally, P(A 1 and and A k ) = P(A 1 ) P(A k ) You toss a coin twice, what is the probability of getting two tails in a row? P(T on the first toss) P(T on the second toss) = = 1 4 Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
49 Independence Question A recent Gallup poll suggests that 25.5% of Texans are uninsured as of June Assuming that the uninsured rate stayed constant, what is the probability that two randomly selected Texans are both uninsured? (a) (b) (c) (d) ( ) poll/ / uninsured-rate-stable-across-states-far-2012.aspx Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
50 Independence Question A recent Gallup poll suggests that 25.5% of Texans are uninsured as of June Assuming that the uninsured rate stayed constant, what is the probability that two randomly selected Texans are both uninsured? (a) (b) (c) (d) ( ) poll/ / uninsured-rate-stable-across-states-far-2012.aspx Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
51 Recap 1 Announcements 2 Probability Randomness Defining probability Law of large numbers Disjoint and non-disjoint outcomes Probability distributions Independence Recap 3 Marginal, joint, conditional Statistics 101
52 Recap 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? Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
53 Recap Disjoint vs. complementary Do the sum of probabilities of two disjoint events always add up to 1? Not necessarily, there may be more than 2 events in the sample space, e.g. party affiliation. Do the sum of probabilities of two complementary events always add up to 1? Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
54 Recap Disjoint vs. complementary Do the sum of probabilities of two disjoint events always add up to 1? Not necessarily, there may be more than 2 events in the sample space, e.g. party affiliation. Do the sum of probabilities of two complementary events always add up to 1? Yes, that s the definition of complementary, e.g. heads and tails. Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
55 Recap Putting everything together... If we were to randomly select 5 Texans, what is the probability that at least one is uninsured? If we were to randomly select 5 Texans, the sample space for the number of Texans who are uninsured would be: S = {0, 1, 2, 3, 4, 5} We are interested in instances where at least one person is uninsured: 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 (Thomas Leininger) U2 - L1: Probability May 21, / 30
56 Recap Putting everything together... Since the probability of the sample space must add up to 1: Prob(at least 1 uninsured) = 1 Prob(none uninsured) Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
57 Recap Putting everything together... Since the probability of the sample space must add up to 1: Prob(at least 1 uninsured) = 1 Prob(none uninsured) = 1 [( ) 5 ] Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
58 Recap Putting everything together... Since the probability of the sample space must add up to 1: Prob(at least 1 uninsured) = 1 Prob(none uninsured) = 1 [( ) 5 ] = Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
59 Recap Putting everything together... Since the probability of the sample space must add up to 1: Prob(at least 1 uninsured) = 1 Prob(none uninsured) = 1 [( ) 5 ] = = Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
60 Recap Putting everything together... Since the probability of the sample space must add up to 1: Prob(at least 1 uninsured) = 1 Prob(none uninsured) = 1 [( ) 5 ] = = = 0.77 At least 1 P(at least one) = 1 P(none) Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
61 Recap Question Roughly 20% of Duke undergraduates are vegetarian or vegan (estimate based on a past class survey). What is the probability that, among a random sample of 3 Duke undergraduates, at least one is vegetarian or vegan? (a) (b) (c) (d) (e) Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
62 Recap Question Roughly 20% of Duke undergraduates are vegetarian or vegan (estimate based on a past class survey). What is the probability that, among a random sample of 3 Duke undergraduates, at least one is vegetarian or vegan? (a) (b) (c) (d) (e) P(at least 1 from veg) = 1 P(none veg) = 1 (1 0.2) 3 = = = Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
63 Marginal, joint, conditional 1 Announcements 2 Probability Randomness Defining probability Law of large numbers Disjoint and non-disjoint outcomes Probability distributions Independence Recap 3 Marginal, joint, conditional Statistics 101
64 Marginal, joint, conditional Relapse Researchers randomly assigned 72 chronic users of cocaine into three groups: desipramine (antidepressant), lithium (standard treatment for cocaine) and placebo. Results of the study are summarized below. no relapse relapse total desipramine lithium placebo total srp/ stats/ 2 way tbl 1.htm Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
65 Marginal, joint, conditional Marginal probability What is the probability that a patient relapsed? no relapse relapse total desipramine lithium placebo total Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
66 Marginal, joint, conditional Marginal probability What is the probability that a patient relapsed? P(relapsed) = no relapse relapse total desipramine lithium placebo total Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
67 Marginal, joint, conditional Joint probability What is the probability that a patient received the the antidepressant (desipramine) and relapsed? no relapse relapse total desipramine lithium placebo total Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
68 Marginal, joint, conditional Joint probability What is the probability that a patient received the the antidepressant (desipramine) and relapsed? no relapse relapse total desipramine lithium placebo total P(relapsed and desipramine) = Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
69 Marginal, joint, conditional Conditional probability If we know that a patient received the antidepressant (desipramine), what is the probability that they relapsed? no relapse relapse total desipramine lithium placebo total Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
70 Marginal, joint, conditional Conditional probability If we know that a patient received the antidepressant (desipramine), what is the probability that they relapsed? no relapse relapse total desipramine lithium placebo total P(relapsed desipramine) = Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
71 Marginal, joint, conditional Conditional probability If we know that a patient received the antidepressant (desipramine), what is the probability that they relapsed? no relapse relapse total desipramine lithium placebo total P(relapsed desipramine) = P(relapsed lithium) = P(relapsed placebo) = Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
72 Marginal, joint, conditional Conditional probability If we know that a patient relapsed, what is the probability that they received the antidepressant (desipramine)? no relapse relapse total desipramine lithium placebo total Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
73 Marginal, joint, conditional Conditional probability If we know that a patient relapsed, what is the probability that they received the antidepressant (desipramine)? no relapse relapse total desipramine lithium placebo total P(desipramine relapsed) = Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
74 Marginal, joint, conditional Conditional probability If we know that a patient relapsed, what is the probability that they received the antidepressant (desipramine)? no relapse relapse total desipramine lithium placebo total P(desipramine relapsed) = P(lithium relapsed) = P(placebo relapsed) = Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
75 Marginal, joint, conditional What if we don t have counts to create a contingency table with counts? Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
76 Marginal, joint, conditional What if we don t have counts to create a contingency table with counts?... next time Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, / 30
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