Econ 240ASummary: Week OneProbabilityOutlineOutline continuedWhy study probability?Cont.ConceptsConceptUncertainty in LifePowerPoint PresentationSlide 12Slide 13Slide 14Slide 15Properties of probabilitiesFlipping a coin twice: 4 elementary outcomesThrowing Two Dice, 36 elementary outcomesSlide 19Combining Elementary Outcomes Into EventsSlide 21Slide 22Operations on eventsProbability statementsSlide 25Slide 26Slide 27ProblemSlide 29Conditional ProbabilitySlide 31Slide 32Independence of two eventsConceptProblem 6.28Problem (Cont.)Slide 37Slide 38Slide 39Slide 40Problem 6.61Slide 42Slide 43Probability 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 events6Why 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 trials7Cont.•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 observations8Concepts•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, six9Concept •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) = 1/610Uncertainty in Life•Demography–Death rates –Marriage–divorce11Uncertainty in Life: US (CDC)1213Probability of First Marriage by Age, Women: US (CDC)14Cohabitation: The Path to Marriage?: US(CDC)15Race/ethnicity Affects Duration of First MarriageProperties of probabilities•Nonnegative–example: p(h) •probabilities of elementary events sum to one–example p(h) + p(t) = 1017Flipping a coin twice: 4 elementary outcomesheadstailsheadstailsheadstailsh, hh, tt, ht, t18Throwing Two Dice, 36 elementary outcomes19Larry Gonick and Woollcott Smith,The Cartoon Guideto Statistics20Combining 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 three21Event: white die equals one is the bottom rowEvent: red die equals one is the right hand column22Event: 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 25Probability 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 one)11( RWp 27Probability 2 diceadd to 6 or add to 3 are mutually exclusive eventsProbability of not rolling snake eyesis easier to calculateas one minus the probability of rolling snake eyesProblem•What is the probability of rolling at least one six in two rolls of a single die?–At least one six is one or two sixes–easier to calculate the probability of rolling zero sixes: (5/36 + 5/36 + 5/36 + 5/36 + 5/36) = 25/36 –and then calculate the probability of rolling at least one six: 1- 25/36 = 11/36)'6(1)'66( szeropstwoonep 29123456123456Probability tree2 rolls of a die:36 elementary outcomes, of which 11 involve one or more sixes30Conditional Probability•Example: in rolling two dice, what is the probability of getting a red one given that you rolled a white one?–P(R1/W1) ?31In rolling two dice, what is the probability of getting a red one giventhat you rolled a white one?Conditional Probability•Example: in rolling two dice, what is the probability of getting a red one given that you rolled a white one?–P(R1/W1) ?)6/1/()36/1()1(/)11()1/1( WpWRpWRp Independence of two events•p(A/B) = p(A)–i.e. if event A is not conditional on event B–then )(*)( BpApBAp 34Concept•Bernoulli Trial–two outcomes, e.g. success or failure–successive independent trials–probability of success is the same in each trial•Example: flipping a coin multiple timesProblem 6.28cash Credit card Debit card<$20 0.09 0.03 0.04$20-$100 0.05 0.21 0.18>$100 0.03 0.23 0.14Distribution of a retail store purchases classified by amountand method of payment36Problem (Cont.)•A. What proportion of purchases was paid by debit card?•B. Find the probability a credit card purchase was over $100•C. Determine the proportion of purchases made by credit card or debit cardProblem 6.28cash Credit card Debit card<$20 0.09 0.03 0.04$20-$100 0.05 0.21 0.18>$100 0.03 0.23 0.14Total 0.17 0.47 0.3638Problem (Cont.)•A. What proportion of purchases was paid by debit card? 0.36•B. Find the probability a credit card purchase was over $100•C. Determine the proportion of purchases made by credit card or debit card39Problem (Cont.)•A. What proportion of purchases was paid by debit card?•B. Find the probability a credit card purchase was over $100 p(>$100/credit card) = 0.23/0.47 = 0.489•C. Determine the proportion of purchases made by credit card or debit card40Problem (Cont.)•A. What proportion of purchases was paid by debit card?•B. Find the probability a credit card purchase was over $100•C. Determine the proportion of purchases made by credit card or
View Full Document
Unlocking...