Econ 240ASummary: Week OneProbabilityOutlineOutline continuedPerspectives About ProbabilityWhy study probability?Cont.Uncertainty in LifeSlide 10Slide 11Slide 12Slide 13Slide 14ConceptsAxiomatic Basis or ConceptsAxiomatic Basis or Concepts IIAxiomatic Basis or Concepts IIIConceptProperties of probabilitiesAnother Example: Toss Two CoinsFlipping a coin twice: 4 elementary outcomesSlide 23Slide 24Slide 25Throwing Two Dice, 36 elementary outcomesSlide 27Combining Elementary Outcomes Into EventsSlide 29Slide 30Slide 31Operations on eventsProbability statementsSlide 34Slide 35Slide 36ProblemSlide 38Conditional ProbabilitySlide 40Slide 41Independence of two eventsConceptProblem 6.28Problem (Cont.)Slide 46Slide 47Slide 48Slide 49Problem 6.61Slide 51Slide 52Probability of a heart attack in the next ten yearsSummary: Probability Rules1Econ 240APower Three2Summary: Week One•Descriptive Statistics–measures of central tendency–measures of dispersion•Distributions of observation values–Histograms: frequency(number) Vs. value•Exploratory data Analysis–stem and leaf diagram–box and whiskers diagram3Probability The GamblerKenny Rogers20 Great Years4Outline•Why study probability?•Random Experiments and Elementary Outcomes•Notion of a fair game•Properties of probabilities•Combining elementary outcomes into events•probability statements•probability trees5Outline continued•conditional probability•independence of two events6Perspectives About Probability•Logical Discipline (like economics)–Axiomatic: conclusions follow from assumptions•Easier to Understand with Examples–I will use words, symbols and pictures•Test Your Understanding By Working Problems7Why study probability?•Understand the concept behind a random sample and why sampling is important–independence of two or more events•understand a Bernoulli event–example; flipping a coin•understand an experiment or a sequence of independent Bernoulli trials8Cont.•Understand the derivation of the binomial distribution, i.e. the distribution of the number of successes, k, in n Bernoulli trials•understand the normal distribution as a continuous approximation to the discrete binomial•understand the likelihood function, i.e. the probability of a random sample of observations9Uncertainty in Life•Demography–Death rates –Marriage–divorce10Uncertainty in Life: US (CDC)1112Probability of First Marriage by Age, Women: US (CDC)13Cohabitation: The Path to Marriage?: US(CDC)14Race/ethnicity Affects Duration of First Marriage15Concepts•Random experiments•Elementary outcomes•example: flipping a coin is a random experiment–the elementary outcomes are heads, tails•example: throwing a die is a random experiment–the elementary outcomes are one, two, three, four, five, six16Axiomatic Basis or Concepts•Elementary outcomes have non-negative probabilities: P(H)>=0, P(T)>=0•The sum of the probabilities over all elementary outcomes equals one: P(H) + P(T) = 1HHTFlip a coin17Axiomatic Basis or Concepts II•The probability of two mutually exculsive events is zero: P(H and T) = P(H^T) = 0•The probability of one outcome or the other is the sum of the probabilities of each minus any double counting: P(H or T) = P(H U T) = P(H) + P(T) – P(H^T)•The probability of the event not happening is one minus the probability of the event happening:)(1)()( HPTPHP 18Axiomatic Basis or Concepts III•Conditional probability of heads given tails equals the joint probability divided by the probability of tails: P(H/T) = P(H^T)/P(T)19Concept •A fair game•example: the probability of heads, p(h), equals the probability of tails, p(t): p(h) = p(t) =1/2•example: the probability of any face of the die is the same, p(one) = p(two) = p(three) = p(four) =p(five) = p(six) = 1/6Properties of probabilities•Nonnegative–example: p(h) •probabilities of elementary events sum to one–example p(h) + p(t) = 1021Another Example: Toss Two Coins H1T1H2T2H, HH, TH2T2T, HT, T22Flipping a coin twice: 4 elementary outcomesheadstailsheadstailsheadstailsh, hh, tt, ht, t23Axiomatic Basis or Concepts•Elementary outcomes have non-negative probabilities: P(H, H)>=0, P(H, T)>=0, P(T, H)>=0, P(T, T) >=0•The sum of the probabilities over all elementary outcomes equals one: P(H, H) + P(H, T) + P(T, H) + P(T, T) = 1•The probability of two mutually exculsive events is zero: P[(H, H)^(H, T)] = 0H24Axiomatic Basis or Concepts II•The probability of one outcome or the other is the sum of the probabilities of each minus any double counting: P[(H, H) U (H,T)] = P(H, H) + P(H, T) – P[(H, H)^(H, T)] = P(H, H) + P(H, T)•The probability of the event not happening is one minus the probability of the event happening:),(),(),(),(1),( TTPHTPTHPHHPHHP 25Axiomatic Basis or Concepts III•Conditional probability of heads, heads given heads, tails equals the joint probability divided by the probability of heads, tails: P[(H, H)/(H, T)] = P[(H, H)^(H, T)]/P(H, T)26Throwing Two Dice, 36 elementary outcomes27Larry Gonick and Woollcott Smith,The Cartoon Guideto Statistics28Combining Elementary Outcomes Into Events•Example: throw two dice: event is white die equals one•example: throw two dice and red die equals one•example: throw two dice and the sum is three29Event: white die equals one is the bottom rowEvent: red die equals one is the right hand column30Combining Elementary Outcomes Into Events•Example: throw two dice: event is white die equals one P(W1) =P(W1^R1) + P(W1^R2) + P(W1^R3) + P(W1^R4) + P(W1^R5) + P(W1^R6) = 6/36•example: throw two dice and red die equals one•example: throw two dice and the sum is three31Event: 2 dice sum to three is lower diagonalOperations on events•The event A and the event B both occur:• Either the event A or the event B occurs or both do:•The event A does not occur, i.e.not A: )( BA )( BA AProbability statements•Probability of either event A or event B–if the events are mutually exclusive, then •probability of event B)()()()( BApBpApBAp )(1)( BpBp 0)( BAp 34Probability of a white one or a red one: p(W1) + p(R1) double countsTwo dice are thrown: probability of the white die showing one and the red die showing one36/1136/136/636/6)11()11()1()1()11(36/1)11(RWPRWPRPWPRWPsoRWP36Probability 2 diceadd to 6 or add to 3 are mutually
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