Econ 240ASummary: Week OneProbabilityOutlineOutline continuedWhy study probability?Cont.ConceptsConceptProperties of probabilitiesFlipping a coin twice: 4 elementary outcomesThrowing Two Dice, 36 elementary outcomesSlide 13Combining Elementary Outcomes Into EventsSlide 15Slide 16Operations on eventsProbability statementsSlide 19Slide 20Slide 21ProblemSlide 23Conditional ProbabilitySlide 25Slide 26Independence of two eventsConceptProblem 6.28Problem (Cont.)Slide 31Slide 32Slide 33Slide 34Problem 6.61Slide 36Slide 37Probability of a heart attack in the next ten years1Econ 240APower Three2Summary: Week One•Descriptive Statistics–measures of central tendency–measures of dispersion•Exploratory data Analysis–stem and leaf diagram–box and whiskers diagram3Probability4Outline•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) = 1/6Properties of probabilities•Nonnegative–example: p(h) •probabilities of elementary events sum to one–example p(h) + p(t) = 1011Flipping a coin twice: 4 elementary outcomesheadstailsheadstailsheadstailsh, hh, tt, ht, t12Throwing Two Dice, 36 elementary outcomes13Larry Gonick and Woollcott Smith,The Cartoon Guideto Statistics14Combining 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 three15Event: white die equals one is the bottom rowEvent: red die equals one is the right hand column16Event: 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 19Probability 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 21Probability 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 23123456123456Probability tree2 rolls of a die:36 elementary outcomes, of which 11 involve one or more sixes24Conditional 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) ?25In 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 28Concept•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 payment30Problem (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 0.3632Problem (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 card33Problem (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 card34Problem (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 card–note: credit card and debit card purchases are mutually exclusive–p(credit or debit) = p(credit) + p (debit) = 0.47 + 0.3635Problem 6.61•A survey of middle aged men reveals that 28% of them are balding at the crown of their head. Moreover, it is known that such men have an 18% probability of suffering a heart attack in the next ten years. Men who are not balding in this way have an 11% probability of a heart attack. Find the probability that a middle aged man will
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