Econ 240A Power Three 1 Summary 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 diagram 2 Probability The Gambler Kenny Rogers 20 Great Years 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 Perspectives 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 Problems 6 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 7 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 8 Uncertainty in Life Demography Death rates Marriage divorce 9 Uncertainty in Life US CDC 10 11 Probability of First Marriage by Age Women US CDC 12 Cohabitation The Path to Marriage US CDC 13 Race ethnicity Affects Duration of First Marriage 14 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 15 Axiomatic Basis or Concepts Elementary outcomes have non negative probabilities P H 0 P T 0 H H Flip a coin T The sum of the probabilities over all elementary outcomes equals one P H P T 1 16 Axiomatic 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 P H P T 1 P H 17 Axiomatic 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 18 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 19 Properties of probabilities Nonnegative 0 example p h probabilities of elementary events sum to one example p h p t 1 Another Example Toss Two Coins H2 H H H1 T2 H T H2 T H T1 T2 T T 21 Flipping a coin twice 4 elementary outcomes heads h h tails h t heads heads t h tails tails t t 22 Axiomatic Basis or Concepts Elementary outcomes have non negative probabilities P H H 0 P H T 0 H 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 0 23 Axiomatic 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 P H H 1 P H H P H T P T H P T T 24 Axiomatic 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 25 Throwing Two Dice 36 elementary outcomes 26 Larry Gonick and Woollcott Smith The Cartoon Guide to Statistics 27 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 28 Event white die equals one is the bottom row Event red die equals one is the right hand column 29 Combining 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 three 30 Event 2 dice sum to three is lower diagonal 31 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 34 Two dice are thrown probability of the white die showing one and the red die showing one P W 1 R1 1 36 so P W 1 R1 P W 1 P R1 P W 1 R1 P W 1 R1 6 36 6 36 1 36 11 36 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 36 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 38 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 39 In rolling two dice what is the probability of getting a red one given that you rolled a white one …
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