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Chapter 15 Probability Models The Equally Likely Approach (also called the Classical Approach)

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Assigning Probabilities zIf an experiment has N outcomes, then each outcome has probability 1/N of occurring zIf an event A 1 has n 1 outcomes, then P(A 1 ) = n 1 /N

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Dice You toss two dice. What is the probability of the outcomes summing to 5? There are 36 possible outcomes in S, all equally likely (given fair dice). Thus, the probability of any one of them is 1/36. P(the roll of two dice sums to 5) = P(1,4) + P(2,3) + P(3,2) + P(4,1) = 4 / 36 = 0.111 This is S: {(1,1), (1,2), (1,3), ……etc.}

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We Need Efficient Methods for Counting Outcomes

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Counting in “Either-Or” Situations NCAA Basketball Tournament, 68 teams: how many ways can the “bracket” be filled out? 1.How many games? 2.2 choices for each game 3.Number of ways to fill out the bracket: 2 67 = 1.5 × 10 20 Earth pop. about 6 billion; everyone fills out 100 million different brackets Chances of getting all games correct is about 1 in 1,000

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Counting Example zPollsters minimize lead-in effect by rearranging the order of the questions on a survey zIf Gallup has a 5-question survey, how many different versions of the survey are required if all possible arrangements of the questions are included?

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Solution zThere are 5 possible choices for the first question, 4 remaining questions for the second question, 3 choices for the third question, 2 choices for the fourth question, and 1 choice for the fifth question. zThe number of possible arrangements is therefore 5 4 3 2 1 = 120

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Efficient Methods for Counting Outcomes zFactorial Notation: n!=1 2 … n zExamples 1!=1; 2!=1 2=2; 3!= 1 2 3=6; 4!=24; 5!=120; zSpecial definition: 0!=1

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Factorials with calculators and Excel zCalculator: non-graphing: x ! (second function) graphing: bottom p. 9 T I Calculator Commands (math button) zExcel: Insert function: Math and Trig category, FACT function

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Factorial Examples z20! = 2.43 x 10 18 z1,000,000 seconds? zAbout 11.5 days z1,000,000,000 seconds? zAbout 31 years z31 years = 10 9 seconds z10 18 = 10 9 x 10 9 z20! is roughly the age (according to some) of the universe in seconds

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Permutations A B C D E zHow many ways can we choose 2 letters from the above 5, without replacement, when the order in which we choose the letters is important? z5 4 = 20

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Permutations (cont.)

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Permutations with calculator and Excel zCalculator non-graphing: nPr zGraphing p. 9 of T I Calculator Commands (math button) zExcel Insert function: Statistical, Permut

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Combinations A B C D E zHow many ways can we choose 2 letters from the above 5, without replacement, when the order in which we choose the letters is not important? z5 4 = 20 when order important Divide by 2: (5 4)/2 = 10 ways

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Combinations (cont.)

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ST 305 Powerball Lottery From the numbers 1 through 20, choose 6 different numbers. Write them on a piece of paper.

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Chances of Winning?

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Example: Illinois State Lottery

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North Carolina Powerball Lottery Prior to Jan. 1, 2009 After Jan. 1, 2009 Most recent change: powerball number is from 1 to 35 http://www.nc-educationlottery.org/faq_powerball.aspx#43

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The Forrest Gump Visualization of Your Lottery Chances zHow large is 195,249,054? z$1 bill and $100 bill both 6” in length z10,560 bills = 1 mile zLet’s start with 195,249,053 $1 bills and one $100 bill … z… and take a long walk, putting down bills end-to-end as we go

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Raleigh to Ft. Lauderdale… … still plenty of bills remaining, so continue from …

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… Ft. Lauderdale to San Diego … still plenty of bills remaining, so continue from…

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… San Diego to Seattle

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… still plenty of bills remaining, so continue from … … Seattle to New York

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… still plenty of bills remaining, so … … New York back to Raleigh

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Go around again! Lay a second path of bills Still have ~ 5,000 bills left!!

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Chances of Winning NC Powerball Lottery? zRemember: one of the bills you put down is a $100 bill; all others are $1 bills. zPut on a blindfold and begin walking along the trail of bills. zYour chance of winning the lottery is the same as your chance of selecting the single $100 bill if you stop at a random location along the trail and pick up a bill.

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More Changes After Jan. 1, 2009 After Jan. 1, 2012 z http://www.nc- educationlottery.org/pow erball_how-to-play.aspx http://www.nc- educationlottery.org/pow erball_how-to-play.aspx

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Virginia State Lottery

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