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Discrete Probability Distributions To accompany Hawkes lesson 5.1 Original content by D.R.S.

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Examples of Probability Distributions Rolling a single dieTotal of rolling two dice ValueProb.ValueProb. 21/3685/36 32/3694/36 43/36103/36 54/36112/36 65/36121/36 76/36Total1 ValueProbability 11/6 2 3 4 5 6 Total1 (Note that it’s a two-column chart but we had to typeset it this way to fit it onto the slide.)

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Example of a Probability Distribution http://en.wikipedia.org/wiki/Poker_probability http://en.wikipedia.org/wiki/Poker_probability Draw this 5-card poker handProbability Royal Flush 0.000154% Straight Flush (not including Royal Flush) 0.00139% Four of a Kind 0.0240% Full House 0.144% Flush (not including Royal Flush or Straight Flush) 0.197% Straight (not including Royal Flush or Straight Flush) 0.392% Three of a Kind 2.11% Two Pair 4.75% One Pair42.3% Something that’s not special at all50.1% Total (inexact, due to rounding)100%

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Exact fractions avoid rounding errors (but is it useful to readers?) Draw this 5-card poker handProbability Royal Flush 4 / 2,598,960 Straight Flush (not including Royal Flush) 36 / 2,598,960 Four of a Kind 624 / 2,598,960 Full House 3,744 / 2,598,960 Flush (not including Royal Flush or Straight Flush) 5,108 / 2,598,960 Straight (not including Royal Flush or Straight Flush) 10,200 / 2,598,960 Three of a Kind 54,912 / 2,598,960 Two Pair 123,552 / 2,598,960 One Pair1,098,240 / 2,598,960 Something that’s not special at all1,302,540 / 2,598,960 Total (exact, precise, beautiful fractions)2,598,600 / 2,598,600

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Example of a probability distribution “How effective is Treatment X?” OutcomeProbability The patient is cured.85% The patient’s condition improves.10% There is no apparent effect. 4% The patient’s condition deteriorates. 1%

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A Random Variable The value of “x” is determined by chance Or “could be” determined by chance As far as we know, it’s “random”, “by chance” The important thing: it’s some value we get in a single trial of a probability experiment It’s what we’re measuring

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Discrete vs. Continuous Discrete A countable number of values “Red”, “Yellow”, “Green” 2 of diamonds, 2 of hearts, … etc. 1, 2, 3, 4, 5, 6 rolled on a die Continuous All real numbers in some interval An age between 10 and 80 (10.000000 and 80.000000) A dollar amount A height or weight

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Discrete is our focus for now Discrete A countable number of values (outcomes) “Red”, “Yellow”, “Green” “Improved”, “Worsened” 2 of diamonds, 2 of hearts, … etc. What poker hand you draw. 1, 2, 3, 4, 5, 6 rolled on a die Total dots in rolling two dice Continuous Will talk about continuous probability distributions in future chapters.

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Start with a frequency distribution General layout A specific made-up example How many children live here? Number of households 050 1100 2150 380 440 520 6 or more10 Total responses450 OutcomeCount of occurrences

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Include a Relative Frequency column General layout A specific simple example # of child ren Number of households Relative Frequency 0500.108 11100.239 21500.326 3800.174 4400.087 5200.043 6+100.022 Total4601.000 OutcomeCount of occur- rences Relative Frequency =count ÷ total

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You can drop the count column General layout A specific simple example # of childrenRelative Frequency 00.108 10.239 20.326 30.174 40.087 50.043 6+0.022 Total1.000 OutcomeRelative Frequency =count ÷ total

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Sum MUST BE EXACTLY 1 !!! In every Probability Distribution, the total of the probabilities must always, every time, without exception, be exactly 1.00000000000. – In some cases, it might be off a hair because of rounding, like 0.999 for example. – If you can maintain exact fractions, this rounding problem won’t happen.

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Answer Probability Questions What is the probability … …that a randomly selected household has exactly 3 children? …that a randomly selected household has children? … that a randomly selected household has fewer than 3 children? … no more than 3 children? A specific simple example # of childrenRelative Frequency 00.108 10.239 20.326 30.174 40.087 50.043 6+0.022 Total1.000

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Answer Probability Questions Referring to the Poker probabilities table “What is the probability of drawing a Four of a Kind hand?” “What is the probability of drawing a Three of a Kind or better?” “What is the probability of drawing something worse than Three of a Kind?” “What is the probability of a One Pair hand twice in a row? (after replace & reshuffle?)”

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Theoretical Probabilities Rolling one die ValueProbability 11/6 2 3 4 5 6 Total1 Total of rolling two dice ValueProb.ValueProb. 21/3685/36 32/3694/36 43/36103/36 54/36112/36 65/36121/36 76/36Total1

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Tossing coin and counting Heads One Coin How many headsProbability 01 / 2 1 Total1 Four Coins How many headsProbability 01/16 14/16 26/16 34/16 41/16 Total1

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Tossing coin and counting Heads How did we get this?Four Coins How many headsProbability 01/16 13/16 26/16 33/16 41/16 Total1 Could try to list the entire sample space: TTTT, TTTH, TTHT, TTHH, THTT, etc. Could use a tree diagram to get the sample space. Could use nCr combinations. We will formally study The Binomial Distribution soon.

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Graphical Representation Histogram, for exampleFour Coins How many headsProbability 01/16 14/16 26/16 34/16 41/16 Total1 6/16 4/16 1/16 0 1 2 3 4 heads Probability

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Shape of the distribution Histogram, for exampleDistribution shapes matter! 6/16 3/16 1/16 0 1 2 3 4 heads Probability This one is a bell-shaped distribution Rolling a single die: its graph is a uniform distribution Other distribution shapes can happen, too

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Remember the Structure Required features The left column lists the sample space outcomes. The right column has the probability of each of the outcomes. The probabilities in the right column must sum to exactly 1.0000000000000000000. Example of a Discrete Probability Distribution # of childrenRelative Frequency 00.108 10.239 20.326 30.174 40.087 50.043 6+0.022 Total1.000

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The Formulas

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TI-84 Calculations Put the outcomes into a TI-84 List (we’ll use L 1 ) Put the corresponding probabilities into another TI-84 List (we’ll use L 2 ) 1-Var Stats L 1, L 2 You can type fractions into the lists, too!

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Practice Calculations Rolling one die ValueProbability 11/6 2 3 4 5 6 Total1 Statistics

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Practice Calculations StatisticsTotal of rolling two dice ValueProb.ValueProb. 21/3685/36 32/3694/36 43/36103/36 54/36112/36 65/36121/36 76/36Total1

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Practice Calculations One Coin How many headsProbability 01 / 2 1 Total1 Statistics

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Practice Calculations StatisticsFour Coins How many headsProbability 01/16 14/16 26/16 34/16 41/16 Total1

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Expected Value Probability Distribution with THREE columns – Event – Probability of the event – Value of the event (sometimes same as the event) Examples: – Games of chance – Insurance payoffs – Business decisions

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Expected Value Problems The Situation 1000 raffle tickets are sold You pay $5 to buy a ticket First prize is $2,000 Second prize is $1,000 Two third prizes, each $500 Three more get $100 each The other ____ are losers. What is the “expected value” of your ticket? The Discrete Probability Distr. OutcomeNet ValueProbability Win first prize $1,9951/1000 Win second prize $9951/1000 Win third prize $4952/1000 Win fourth prize $953/1000 Loser$ -5993/1000 Total1000/1000

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Expected Value Problems Statistics The mean of this probability is $ - 0.70, a negative value. This is also called “Expected Value”. Interpretation: “On the average, I’m going to end up losing 70 cents by investing in this raffle ticket.” The Discrete Probability Distr. OutcomeNet ValueProbability Win first prize $1,9951/1000 Win second prize $9951/1000 Win third prize $4952/1000 Win fourth prize $953/1000 Loser$ -5993/1000 Total1000/1000

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Expected Value Problems Another way to do it Use only the prize values. The expected value is the mean of the probability distribution which is $4.30 Then at the end, subtract the $5 cost of a ticket, once. Result is the same, an expected value = $ -0.70 The Discrete Probability Distr. OutcomeNet ValueProbability Win first prize $2,0001/1000 Win second prize $1,0001/1000 Win third prize $5002/1000 Win fourth prize $1003/1000 Loser$ 0993/1000 Total1000/1000

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Expected Value Problems The Situation We’re the insurance company. We sell an auto policy for $500 for 6 months coverage on a $20,000 car. The deductible is $200 What is the “expected value” – that is, profit – to us, the insurance company? The Discrete Probability Distr. OutcomeNet ValueProbability No claims filed _______ An $800 fender bender 0.004 An $8,000 accident 0.002 A wreck, it’s totaled 0.002

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An Observation The mean of a probability distribution is really the same as the weighted mean we have seen. Recall that GPA is a classic instance of weighted mean – Grades are the values – Course credits are the weights Think about the raffle example – Prizes are the values – Probabilities of the prizes are the weights

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