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Chapter 15 Probability Models

Chapter 15 Probability Models. The Equally Likely Approach (also called the Classical Approach). Assigning Probabilities. If an experiment has N outcomes, then each outcome has probability 1/N of occurring If an event A 1 has n 1 outcomes, then P(A 1 ) = n 1 /N. Dice

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Chapter 15 Probability Models

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

  2. Assigning Probabilities • If an experiment has N outcomes, then each outcome has probability 1/N of occurring • If an event A1 has n1 outcomes, then P(A1) = n1/N

  3. Dice You toss two dice. What is the probability of the outcomes summing to 5? This isS: {(1,1), (1,2), (1,3), ……etc.} 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

  4. We Need Efficient Methods for Counting Outcomes

  5. Counting in “Either-Or” Situations • NCAA Basketball Tournament, 68 teams: how many ways can the “bracket” be filled out? • How many games? • 2 choices for each game • Number of ways to fill out the bracket: 267 = 1.5 × 1020 • Earth pop. about 6 billion; everyone fills out 100 million different brackets • Chances of getting all games correct is about 1 in 1,000

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

  7. Solution • There 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. • The number of possible arrangements is therefore 5  4  3  2  1 = 120

  8. Efficient Methods for Counting Outcomes • Factorial Notation: n!=12 … n • Examples 1!=1; 2!=12=2; 3!= 123=6; 4!=24; 5!=120; • Special definition: 0!=1

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

  10. Factorial Examples • 20! = 2.43 x 1018 • 1,000,000 seconds? • About 11.5 days • 1,000,000,000 seconds? • About 31 years • 31 years = 109 seconds • 1018 = 109 x 109 • 20! is roughly the age of the universe in seconds

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

  12. Permutations (cont.)

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

  14. Combinations A B C D E • How 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? • 5 4 = 20 when order important • Divide by 2: (5  4)/2 = 10 ways

  15. Combinations (cont.)

  16. Example: Illinois State Lottery

  17. North Carolina Powerball Lottery Prior to Jan. 1, 2009 After Jan. 1, 2009

  18. The Forrest Gump Visualization of Your Lottery Chances • How large is 195,249,054? • $1 bill and $100 bill both 6” in length • 10,560 bills = 1 mile • Let’s start with 195,249,053 $1 bills and one $100 bill … • … and take a long walk, putting down bills end-to-end as we go

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

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

  21. … San Diego to Seattle … still plenty of bills remaining, so continue from …

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

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

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

  25. Chances of Winning NC Powerball Lottery? • Remember: one of the bills you put down is a $100 bill; all others are $1 bills. • Put on a blindfold and begin walking along the trail of bills. • Your 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 .

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

  27. Virginia State Lottery

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