Econ 240A Power Three 1 Summary Week One Descriptive Statistics measures of central tendency measures of dispersion Exploratory data Analysis stem and leaf diagram box and whiskers diagram 2 Probability 3 Outline Why study probability Random Experiments and Elementary Outcomes Notion of a fair game Properties of probabilities Combining elementary outcomes into events probability statements probability trees 4 Outline continued conditional probability independence of two events 5 Why 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 trials 6 Cont 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 observations 7 Concepts 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 six 8 Concept 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 6 9 Properties of probabilities example p h 0 Nonnegative probabilities of elementary events sum to one example p h p t 1 Flipping a coin twice 4 elementary outcomes heads h h tails h t heads heads t h tails tails t t 11 Throwing Two Dice 36 elementary outcomes 12 Larry Gonick and Woollcott Smith The Cartoon Guide to Statistics 13 Combining 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 three 14 Event white die equals one is the bottom row Event red die equals one is the right hand column 15 Event 2 dice sum to three is lower diagonal 16 Operations on events The event A and the event B both occur A B Either the event A or the event B occurs or both do A B The event A does not occur i e not A A Probability statements Probability of either event A or event B p A B p A p B p A B if the events are mutually exclusive then p A B 0 probability of event B p B 1 p B Probability of a white one or a red one p W1 p R1 double counts 19 Two dice are thrown probability of the white die showing one and the red die showing one p W 1 R1 Probability 2 dice add to 6 or add to 3 are mutually exclusive events Probability of not rolling snake eyes is easier to calculate as one minus the probability of rolling snake eyes 21 Problem 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 p one6 two6 s 1 p zero 6 s 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 Probability tree 1 2 1 3 4 2 5 3 6 4 5 6 2 rolls of a die 36 elementary outcomes of which 11 involve one or more sixes 23 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 24 In rolling two dice what is the probability of getting a red one given that you rolled a white one 25 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 p R1 W 1 p R1 W 1 p W 1 1 36 1 6 Independence of two events p A B p A i e if event A is not conditional on event B then p A B p A p B Concept 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 times 28 Problem 6 28 cash Credit card Debit card 0 09 0 03 0 04 20 100 0 05 0 21 0 18 100 0 23 0 14 20 0 03 Distribution of a retail store purchases classified by amount and method of payment Problem 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 30 Problem 6 28 cash Credit card Debit card 0 09 0 03 0 04 20 100 0 05 0 21 0 18 100 0 03 0 23 0 14 Total 0 17 0 47 0 36 20 Problem 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 card 32 Problem 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 card 33 Problem 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 36 34 Problem 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 suffer a heart attack in the next ten years 35 P Bald and MA 0 28 Bald Not Bald Middle Aged men 36 P Bald and MA 0 28 P HA Bald and MA 0 18 Bald P HA Not Bald and MA 0 11 Not Bald Middle Aged men 37 Probability of a heart attack in the next ten years P HA P HA and Bald and MA P HA and Not Bald and MA P HA P HA Bald and MA P BALD and MA P HA Not BALD and MA P Not Bald and MA P HA …
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