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Efficient Methods for Counting Outcomes in Probability Models

Learn efficient techniques for counting outcomes in probability models using the classical approach. Understand calculating probabilities, permutations, combinations, and factorial notation. Explore examples and applications in various scenarios.

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Efficient Methods for Counting Outcomes in 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 (according to some) 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. ST 305 Powerball Lottery From the numbers 1 through 20, choose 6 different numbers. Write them on a piece of paper.

  17. Chances of Winning?

  18. Example: Illinois State Lottery

  19. 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

  20. 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

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

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

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

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

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

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

  27. 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 .

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

  29. Virginia State Lottery

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