Slide 1ProbabilityTypes of probabilityTerminologyHeads upLimiting relative frequencyEpistemic probabilityProbabilityNotation we will regularly useFrequency probabilityShorthand for probabilityMarginal and joint probabilityConditional probabilityRelating these probabilitiesIndependent eventsIndependent eventsDependent eventsDependent eventsIndependence in sports“or” rule“or” rule“or” ruleLaw of Total of Probability“or”A common confusionCoin tossExampleExampleExampleExampleExampleLet’s make a dealSlide 33Let’s make a deal revisitedBirthday problemBirthday problemPistols at dawnSlide 38Sports playoffsSports playoffsBayes RuleBayes ruleSlide 43EIA exampleHypothetical table for EIAEIA test: Final TableEIA test: Final TableEIA test: Final tableEIA test: Final answerSensitivity to initial marginal probabilityBinomial distributionBinomial distributionFPP 13-15Probability1ProbabilityWhat statisticians hang their hat onProvides a formal framework from which uncertainty can be quantifiedWhy study probability in an intro stat course?Lay foundations for statistical inference.Train your brain to think in a way that it is not hardwired to doIts quite enjoyable and relaxing2Types of probabilityWhat exactly is probability?There are any number of notions of probability, indicating that probability isn’t a thing but a conceptWe can spend a semester philosophizing about probability if you are interested I can direct you to some books.An unexhausted listLaplacian probabilityHypothetical limiting relative frequency probabilityNomic probabilityFiducial probabilityEpistemic probabilityIn this class we will focus on two of these.3TerminologySample Space: The set (collection) of all possible outcomes that can happen.Event: A single outcome or set of outcomes from a sample spaceProbability Model: A consistent assignment of a probability to each even in the sample spaceDisjoint Events: Two events that have no outcomes in commone and, thus, cannot both occur simultaneouslyVenn Diagrams help visualize the above 4Heads upMathematical notation will become a little more prevalent here. You’ll need to put forth effort wrapping your brain around it.5Limiting relative frequencyMost folks call this the frequentist approach1. Operations: observation, measurement, or selection that can at least hypothetically be repeated an infinite number of times2. Sample space: set of possible outcomes of an operation3. Events: subsets of elements in the sample spaceElements of the sample space (basic outcomes) are equally likely Calculation1. Let S denote the sample space, E ⊂ S denote an event, and |A| denote the size of any set A2. P r(E) ≡ |E|/|S |UpshotPercentage of times an event occurs in repeated realizations of random processes6Epistemic probabilityOften times called “subjective” probabilityThis term is a bit loaded as it can be argued that “objective” probability doesn’t really existHere probability is degree of belief in likelihood of eventBelief is updated or modified in the light of observed information7ProbabilityWhy consider two probabilitiesEach allows different approaches to incorporating probability in an anslysisEach one leads to different types of inference statements.Is one preferable to the other?This really depends on who you ask.There have been (heated) discussions on the appropriateness of both8Notation we will regularly use9Frequency probabilityWe focus first on how to use frequency probability in an analysis and will cover epistemic probability laterSimple motivating exampleThere are 3 red balls and 9 white balls in a hatPick one ball at random out of the hatOnce picked the ball is not replacedThen pick another ball at random out of the hat10Shorthand for probabilityDefine R1 = pick a red ball on the 1st tryDefine R2 = pick a red ball on the 2nd tryDefine W1 = pick a white ball on the 1st tryDefine W2 = pick a white ball on the 2nd tryProbability of picking a red ball on the 1st try isPr(R1) = Probability of picking two red balls in two picks without replacing 1st ball isPr(R1 and R2) = 11Marginal and joint probabilityProbability of a single event is called marginal probabilityExample: Pr(R1)Probability of intersection of two events (both events happening) is called a joint probability Example: Pr(R1 and R2)12Conditional probabilitySay we pick a red ball on the 1st try. The chance we pick a red ball on the 2nd try equals 2/11.Probability that an event given another event occurs is called conditional probabilityShorthand: Pr(R2|R1) = 2/11“Probability that R2 occurs given that R1 occurs.” 13Relating these probabilitiesPr(R1 and R2) = Pr(R1)Pr(R2|R1)6/132 = 3/12(2/11)Joint prob. = marginal prob. times conditional prob.This is always true14Independent eventsReplace 1st ball before picking the 2nd . ThenPr(R1) = 3/12Pr(R2 | R1) = Pr(R2) = 3/12R1 and R2 are called independent events: The occurrence of R1 does not affect the probability of R2. 15Independent eventsWhen events are independent calculating joint probabilities is fairly easyLet events A, B, C, … etc. be independentPr(A and B and C and … etc. ) = Pr(A)Pr(B)Pr(C)Pr(etc.)To get joint probabilities you can simply multiply the marginal probabilitiesWhy does this work?16Dependent eventsNotice that when sampling with out replacement thenPr(R2|R1) = 2/11 ≠ 3/12 = Pr(R2)When the conditional prob. is not equal to the marginal prob. then the events are said to be dependent. The occurrence of R1 affects the probability of R2Here R1 and R2 are dependent events17Dependent eventsWhen events are dependent joint probabilities are harder to computeLet A, B, C, …, etc. be dependent eventsPr(A and B and C and … etc.) = Pr(A|B,C,etc.)Pr(B|C, etc.)Pr(C|etc.)Pr(etc.)To get joint probabilities, you multiply all the conditional probabilities18Independence in sportsBaseball announcers sometimes say, “The batter has not gotten a base hit in the last four times he’s batted. He’s due for a hit now”What is this statement assuming?19“or” ruleThis is an inclusive “or” That is, A or B = A or B or both.Pr(A or B) = Pr(A) + Pr(B) – Pr(A and B)20“or” rulePr(R1 or R2 ) = Pr(R1) + Pr(R2) – Pr(R1 and R2) = Pr(R1) + Pr(R2) –
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