Announcements U nit 2: P robability and distributions L ecture 1: P robability and conditional probability Turn in PS 2 S tatistics 101 We will start Lab 2 today Nicole Dalzell May 19, 2015 Statistics 101 ( Nicole Dalzell ) U2 - L1: Probability May 19, 2015 2 / 1 Probability Randomness Probability Defining probability Random processes Probability There are several possible interpretations of probability but they A random process is a (almost) completely agree on the mathematical rules probability must follow. situation in which we know what outcomes could happen, P ( A ) = Probability of event A 0 ≤ P ( A ) ≤ 1 but we don’t know which Frequentist interpretation: particular outcome will happen. The probability of an outcome is the proportion of times the outcome would occur if we observed the random process an Examples: coin tosses, die infinite number of times. rolls, iTunes shuffle, whether Single main stream school until recently. the stock market goes up or Bayesian interpretation: down tomorrow, etc. A Bayesian interprets probability as a subjective degree of belief: It can be helpful to model a For the same event, two separate people could have differing process as random even if it probabilities. http://www.cnet.com.au/ is not truly random. Largely popularized by revolutionary advance in computational itunes-just-how-random-is-random-339274094.htm technology and methods during the last twenty years. Statistics 101 ( Nicole Dalzell ) U2 - L1: Probability May 19, 2015 3 / 1 Statistics 101 ( Nicole Dalzell ) U2 - L1: Probability May 19, 2015 4 / 1
Probability Law of large numbers Probability Law of large numbers Law of large numbers Participation question Which of the following events would you be most surprised by? Law of large numbers states that as more observations are collected, (a) 3 heads in 10 coin flips the proportion of occurrences with a particular outcome, ˆ p n , (b) 3 heads in 100 coin flips converges to the probability of that outcome, p . (c) 3 heads in 1000 coin flips Statistics 101 ( Nicole Dalzell ) U2 - L1: Probability May 19, 2015 5 / 1 Statistics 101 ( Nicole Dalzell ) U2 - L1: Probability May 19, 2015 6 / 1 Probability Law of large numbers Probability Disjoint and non-disjoint outcomes Law of large numbers vs. law of averages Disjoint and non-disjoint outcomes 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? Disjoint (mutually exclusive) outcomes: Cannot happen at the same time. H H H H H H H H H H ? The outcome of a single coin toss cannot be a head and a tail. A student cannot fail and pass a class. The probability is still 0.5, or there is still a 50% chance that another head will come up on the next toss. A card drawn from a deck cannot be an ace and a queen. P ( H on 11 th toss ) = P ( T on 11 th toss ) = 0 . 5 Non-disjoint outcomes: Can happen at the same time. A student can get an A in Stats and A in Econ in the same The coin is not due for a tail . semester. 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 ( Nicole Dalzell ) U2 - L1: Probability May 19, 2015 7 / 1 Statistics 101 ( Nicole Dalzell ) U2 - L1: Probability May 19, 2015 8 / 1
Probability Disjoint and non-disjoint outcomes Probability Probability distributions Participation question Participation question What is the probability that a randomly sampled student thinks marijuana In a survey, 52% of respondents said they are Democrats. What is the should be legalized or they agree with their parents’ political views? probability that a randomly selected respondent from this sample is a Republican? Parent Politics Legalize MJ No Yes Total (a) 0.48 No 11 40 51 Yes 36 78 114 (b) more than 0.48 Total 47 118 165 (c) less than 0.48 40 + 36 − 78 78 11 (a) (c) (e) 165 165 47 (d) cannot calculate using only the information given 114 + 118 − 78 78 (b) (d) 165 188 Statistics 101 ( Nicole Dalzell ) U2 - L1: Probability May 19, 2015 9 / 1 Statistics 101 ( Nicole Dalzell ) U2 - L1: Probability May 19, 2015 10 / 1 Probability Probability distributions Probability Independence Disjoint vs. complementary Independence Two processes are independent if knowing the outcome of one Do the sum of probabilities of two disjoint events always add up to 1? provides no useful information about the outcome of the other. coin flips are independent card draws (without replacement) are dependent Do the sum of probabilities of two complementary events always add Independence and disjointness do not mean the same thing: up to 1? independent events do not affect each other disjoint (mutually exclusive) events cannot happen at the same time Statistics 101 ( Nicole Dalzell ) U2 - L1: Probability May 19, 2015 11 / 1 Statistics 101 ( Nicole Dalzell ) U2 - L1: Probability May 19, 2015 12 / 1
Probability Independence Probability Independence Participation 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, Checking for independence 28% of Black respondents, and 64% of Hispanic respondents shared this view. Which of the below is true? If P ( A | B ) = P ( A ) , then A and B are independent. Opinion on gun ownership and race ethnicity are most likely (a) complementary (b) mutually exclusive (c) independent (d) dependent (e) disjoint http://www.surveyusa.com/client/PollReport.aspx?g=a5f460ef-bba9-484b-8579-1101ea26421b Statistics 101 ( Nicole Dalzell ) U2 - L1: Probability May 19, 2015 13 / 1 Statistics 101 ( Nicole Dalzell ) U2 - L1: Probability May 19, 2015 14 / 1 Probability Independence Probability Independence Evaluating dependence based on sample data Participation question Conditional probabilities calculated based on sample data suggest dependence → conduct a hypothesis test to determine if the A 2012 Gallup poll suggests that 19.4% of North Carolinians don’t have health observed difference is unlikely to have happened by chance. insurance. Assuming that the uninsured rate stayed constant, what is the probability that two randomly selected North Carolinians are both uninsured? We have seen that P(protects citizens | White) = 0.67 and P(protects citizens (a) 19 . 4 2 | Hispanic) = 0.64. Under which condition would you be more convinced of (b) 0 . 194 2 a real difference between the proportions of Whites and Hispanics who think gun widespread gun ownership protects citizens? n = 500 or n = 50 , 000 (c) 0 . 194 × 2 (d) ( 1 − 0 . 194 ) 2 Large observed difference → hypothesis test will likely be significant. http://www.gallup.com/poll/125066/ Small observed difference, and State-States.aspx?ref=interactive n large → hypothesis test may be significant. n small → then hypothesis test will likely not be significant. Statistics 101 ( Nicole Dalzell ) U2 - L1: Probability May 19, 2015 15 / 1 Statistics 101 ( Nicole Dalzell ) U2 - L1: Probability May 19, 2015 16 / 1
Probability Recap Marginal, joint, conditional Relapse Participation question Researchers randomly assigned 72 chronic users of cocaine into In a NC emergency room, 5 patients are waiting to be seen. Assuming that three groups: desipramine (antidepressant), lithium (standard these patients constitute a random sample, what is the probability that at least treatment for cocaine) and placebo. Results of the study are summarized below. one is uninsured? (a) 1 − 0 . 194 × 5 no relapse relapse total (b) 1 − 0 . 194 5 desipramine 10 14 24 (c) 0 . 806 5 lithium 18 6 24 placebo 20 4 24 (d) 1 − 0 . 806 × 5 total 48 24 72 (e) 1 − 0 . 806 5 http://www.oswego.edu/ ∼ srp/stats/2 way tbl 1.htm Statistics 101 ( Nicole Dalzell ) U2 - L1: Probability May 19, 2015 17 / 1 Statistics 101 ( Nicole Dalzell ) U2 - L1: Probability May 19, 2015 18 / 1 Marginal, joint, conditional Marginal, joint, conditional Marginal probability Joint probability What is the probability that a patient received the the antidepressant What is the probability that a patient relapsed? (desipramine) and relapsed? no no relapse relapse total relapse relapse total desipramine 10 14 24 desipramine 10 10 14 24 lithium 18 6 24 lithium 18 6 24 placebo 20 4 24 placebo 20 4 24 total 48 48 24 72 72 total 48 24 72 72 P(relapsed) = 48 72 ≈ 0 . 67 P(relapsed and desipramine) = 10 72 ≈ 0 . 14 Statistics 101 ( Nicole Dalzell ) U2 - L1: Probability May 19, 2015 19 / 1 Statistics 101 ( Nicole Dalzell ) U2 - L1: Probability May 19, 2015 20 / 1
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