Unit 2: Probability And Distributions Lecture 1 .

Unit 2: Probability and distributionsLecture 1: Probability and conditional probabilityStatistics 101Thomas LeiningerMay 21, 2013

AnnouncementsAnnouncementsPS #1 due todayPS #2 assigned (due Friday)Statistics 101 (Thomas Leininger)U2 - L1: ProbabilityMay 21, 20132 / 30

AnnouncementsVisualization of the daywrong,overturnwrong,don't overturnnot wrong,overturnnot wrong,don't overturnNA010203040http:// www.washingtonpost.com/ blogs/ the-fix/ wp/ 2013/ 01/ 22/ de/Statistics 101 (Thomas Leininger)U2 - L1: ProbabilityMay 21, 20133 / 30

ProbabilityRandomnessRandom processesA random process is asituation in which we knowwhat outcomes could happen,but we don’t know whichparticular outcome willhappen.Examples: coin tosses, dierolls, iTunes shuffle, whetherthe stock market goes up ordown tomorrow, etc.It can be helpful to model aprocess as random even if itis not truly random.Statistics 101 (Thomas Leininger)http:// 39274094.htmU2 - L1: ProbabilityMay 21, 20134 / 30

ProbabilityDefining probabilityProbabilityThere are several possible interpretations of probability but they(almost) completely agree on the mathematical rules probabilitymust follow.P (A ) Probability of event A0 P (A ) 1Frequentist interpretation:The probability of an outcome is the proportion of times theoutcome would occur if we observed the random process aninfinite 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 differingprobabilities.Largely popularized by revolutionary advance in computationaltechnology and methods during the last twenty years.Statistics 101 (Thomas Leininger)U2 - L1: ProbabilityMay 21, 20135 / 30

ProbabilityLaw of large numbersQuestionWhich 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 flipsStatistics 101 (Thomas Leininger)U2 - L1: ProbabilityMay 21, 20136 / 30

ProbabilityLaw of large numbersLaw of large numbersLaw of large numbers states that as more observations are collected,the proportion of occurrences with a particular outcome, p̂n ,converges to the probability of that outcome, p.Statistics 101 (Thomas Leininger)U2 - L1: ProbabilityMay 21, 20137 / 30

ProbabilityLaw of large numbersLaw of large numbers (cont.)When tossing a fair coin, if heads comes up on each of the first 10tosses, what do you think the chance is that another head will comeup on the next toss? 0.5, less than 0.5, or more than 0.5?HHHHHHHHHH?The probability is still 0.5, or there is still a 50% chance thatanother head will come up on the next toss.P (H on 11th toss) P (T on 11th toss) 0.5The coin is not due for a tail.The common (mis)understanding of the LLN is that randomprocesses are supposed to compensate for whatever happenedin the past; this is just not true and is also called gambler’s fallacy(or law of averages).Statistics 101 (Thomas Leininger)U2 - L1: ProbabilityMay 21, 20138 / 30

ProbabilityDisjoint and non-disjoint outcomesDisjoint and non-disjoint outcomesDisjoint (mutually exclusive) outcomes: Cannot happen at the sametime.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 samesemester.Statistics 101 (Thomas Leininger)U2 - L1: ProbabilityMay 21, 20139 / 30

ProbabilityDisjoint and non-disjoint outcomesUnion of non-disjoint eventsWhat is the probability of drawing a jack or a red card from a wellshuffled full deck?Figure from http:// www.milefoot.com/ math/ discrete/ counting/ cardfreq.htm .Statistics 101 (Thomas Leininger)U2 - L1: ProbabilityMay 21, 201310 / 30

ProbabilityDisjoint and non-disjoint outcomesRecapGeneral addition ruleP (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 toP (A or B ) P (A ) P (B ).Statistics 101 (Thomas Leininger)U2 - L1: ProbabilityMay 21, 201311 / 30

ProbabilityDisjoint and non-disjoint outcomesQuestionWhat is the probability that a randomly sampled STA 101 studentthinks marijuana should be legalized or they agree with their parents’political views?Legalize MJNoYesTotal(a)(b)(c)(d)(e)Parent PoliticsNoYes1140367847118Total5111416540 36 78165114 118 7816578165781881147* Data from a previous semester.Statistics 101 (Thomas Leininger)U2 - L1: ProbabilityMay 21, 201312 / 30

ProbabilityProbability distributionsProbability distributionsA probability distribution lists all possible events and the probabilitieswith which they occur.The probability distribution for the gender of one kid:EventProbabilityB0.5G0.5Rules for probability distributions:123The events listed must be disjointEach probability must be between 0 and 1The probabilities must total 1The probability distribution for the genders of two kids:Statistics 101 (Thomas Leininger)U2 - L1: ProbabilityMay 21, 201313 / 30

ProbabilityProbability distributionsQuestionIn a survey, 52% of respondents said they are Democrats. What is theprobability that a randomly selected respondent from this sample is aRepublican?(a) 0.48(b) more than 0.48(c) less than 0.48(d) cannot calculate using only the information givenStatistics 101 (Thomas Leininger)U2 - L1: ProbabilityMay 21, 201314 / 30

ProbabilityIndependenceIndependenceTwo processes are independent if knowing the outcome of oneprovides no useful information about the outcome of the other.Knowing that the coin landed on a head on the first toss does notprovide any useful information for determining what the coin willland 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 doesprovide useful information for determining the probability ofdrawing an ace in the second draw. Outcomes of two drawsfrom a deck of cards (without replacement) are dependent.Statistics 101 (Thomas Leininger)U2 - L1: ProbabilityMay 21, 201315 / 30

ProbabilityIndependenceQuestionBetween January 9-12, 2013, SurveyUSA interviewed a random sample of500 NC residents asking them whether they think widespread gun ownershipprotects 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 thisview. 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) disjointhttp:// www.surveyusa.com/ client/ PollReport.aspx?g a5f460ef-bba9-484b-8579-1101ea26421bStatistics 101 (Thomas Leininger)U2 - L1: ProbabilityMay 21, 201316 / 30

ProbabilityIndependenceChecking for independenceIf P (A B ) P (A ), then A and B are independent.Statistics 101 (Thomas Leininger)U2 - L1: ProbabilityMay 21, 201317 / 30

ProbabilityIndependenceDetermining dependence based on sample dataIf conditional probabilities calculated based on sample datasuggest dependence between two variables, the next step is toconduct a hypothesis test to determine if the observed differencebetween the probabilities is likely or unlikely to have happenedby chance.If the observed difference between the conditional probabilities islarge, then the hypothesis test will likely be significant.If the observed difference between the conditional probabilities issmall, and the sample is large as well, the hypothesis test maybe 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) 0.64. Under which condition would you be more convinced ofa real difference between the proportions of Whites and Hispanics who thinkgun widespread gun ownership protects citizens? n 500 or n 50, 000Statistics 101 (Thomas Leininger)U2 - L1: ProbabilityMay 21, 201318 / 30

ProbabilityIndependenceProduct rule for independent eventsP (A and B ) P (A ) P (B )Or more generally, P (A1 and · · · and Ak ) P (A1 ) · · · P (Ak )You toss a coin twice, what is the probability of getting two tails in arow?P (T on the first toss) P (T on the second toss) Statistics 101 (Thomas Leininger)U2 - L1: Probability1 11 2 24May 21, 201319 / 30

ProbabilityIndependenceQuestionA recent Gallup poll suggests that 25.5% of Texans are uninsured as ofJune 2012. Assuming that the uninsured rate stayed constant, what isthe probability that two randomly selected Texans are both uninsured?(a) 25.52(b) 0.2552(c) 0.255 2(d) (1 0.255)2http:// www.gallup.com/ poll/ 156851/ tatistics 101 (Thomas Leininger)U2 - L1: ProbabilityMay 21, 201320 / 30

ProbabilityRecapDisjoint vs. complementaryDo the sum of probabilities of two disjoint events always add up to 1?Do the sum of probabilities of two complementary events always addup to 1?Statistics 101 (Thomas Leininger)U2 - L1: ProbabilityMay 21, 201321 / 30

ProbabilityRecapPutting everything together.If we were to randomly select 5 Texans, what is the probability that atleast one is uninsured?If we were to randomly select 5 Texans, the sample space for thenumber of Texans who are uninsured would be:S {0, 1, 2, 3, 4, 5}We are interested in instances where at least one person isuninsured: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: ProbabilityMay 21, 201322 / 30

ProbabilityRecapPutting everything together.Since the probability of the sample space must add up to 1:Prob (at least 1 uninsured ) 1 Prob (none uninsured ) 1 [(1 0.255)5 ] 1 0.7455 1 0.23 0.77At least 1P (at least one ) 1 P (none )Statistics 101 (Thomas Leininger)U2 - L1: ProbabilityMay 21, 201323 / 30

ProbabilityRecapQuestionRoughly 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 isvegetarian or vegan?(a) 1 0.2 3(b) 1 0.23(c) 0.83(d) 1 0.8 3(e) 1 0.83Statistics 101 (Thomas Leininger)U2 - L1: ProbabilityMay 21, 201324 / 30

Marginal, joint, conditionalRelapseResearchers randomly assigned 72 chronic users of cocaine intothree groups: desipramine (antidepressant), lithium (standardtreatment for cocaine) and placebo. Results of the study aresummarized 8norelapse146424total24242472http:// www.oswego.edu/ srp/ stats/ 2 way tbl 1.htmStatistics 101 (Thomas Leininger)U2 - L1: ProbabilityMay 21, 201325 / 30

Marginal, joint, conditionalMarginal probabilityWhat is the probability that a patient relapsed?desipraminelithiumplacebototalP(relapsed) 4872relapse10182048 48norelapse146424total24242472 72 0.67Statistics 101 (Thomas Leininger)U2 - L1: ProbabilityMay 21, 201326 / 30

Marginal, joint, conditionalJoint probabilityWhat is the probability that a patient received the the antidepressant(desipramine) and relapsed?desipraminelithiumplacebototalrelapse10 10182048P(relapsed and desipramine) Statistics 101 (Thomas Leininger)1072norelapse146424total24242472 72 0.14U2 - L1: ProbabilityMay 21, 201327 / 30

Marginal, joint, conditionalConditional probabilityIf we know that a patient received the antidepressant (desipramine),what is the probability that they relapsed?desipraminelithiumplacebototalP(relapsed desipramine) relapse10 101820481024norelapse146424total24 24242472 0.42P(relapsed lithium) 1824 0.75P(relapsed placebo) 2024 0.83Statistics 101 (Thomas Leininger)U2 - L1: ProbabilityMay 21, 201328 / 30

Marginal, joint, conditionalConditional probabilityIf we know that a patient relapsed, what is the probability that theyreceived the antidepressant (desipramine)?relapse10 10182048 48desipraminelithiumplacebototalP(desipramine relapsed) 1048norelapse146424total24242472 0.2118P(lithium relapsed) 48 0.375P(placebo relapsed) 2048 0.42Statistics 101 (Thomas Leininger)U2 - L1: ProbabilityMay 21, 201329 / 30