1 / 34

4.1 (cont.) Probability Models

4.1 (cont.) 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

vadin
Download Presentation

4.1 (cont.) Probability Models

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 4.1 (cont.) 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. Product Rule for Ordered Pairs • A student wishes to commute to a junior college for 2 years and then commute to a state college for 2 years. Within commuting distance there are 4 junior colleges and 3 state colleges. How many junior college-state college pairs are available to her?

  6. Product Rule for Ordered Pairs • junior colleges: 1, 2, 3, 4 • state colleges a, b, c • possible pairs: (1, a) (1, b) (1, c) (2, a) (2, b) (2, c) (3, a) (3, b) (3, c) (4, a) (4, b) (4, c)

  7. Product Rule for Ordered Pairs • junior colleges: 1, 2, 3, 4 • state colleges a, b, c • possible pairs: (1, a) (1, b) (1, c) (2, a) (2, b) (2, c) (3, a) (3, b) (3, c) (4, a) (4, b) (4, c) 4 junior colleges 3 state colleges total number of possible pairs = 4 x 3 = 12

  8. Product Rule for Ordered Pairs • junior colleges: 1, 2, 3, 4 • state colleges a, b, c • possible pairs: (1, a) (1, b) (1, c) (2, a) (2, b) (2, c) (3, a) (3, b) (3, c) (4, a) (4, b) (4, c) In general, if there are n1 ways to choose the first element of the pair, and n2 ways to choose the second element, then the number of possible pairs is n1n2. Here n1 = 4, n2 = 3.

  9. 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

  10. A state’s automobile license plate begins with a number from 1 to 26, corresponding to the 26 counties in a state. This number is followed by a 5-digit number. How many different license plates can the state issue? • 1,300 • 6,552 • 2,600,000 • 786,240 • 26,000 10 Countdown

  11. 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?

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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

  17. Permutations (cont.)

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

  19. 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

  20. Combinations (cont.)

  21. BUS/ST 350 Powerball Lottery From the numbers 1 through 20, choose 6 different numbers. Write them on a piece of paper.

  22. Chances of Winning?

  23. Example: Illinois State Lottery

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

  25. 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

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

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

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

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

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

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

  32. 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: the chance of selecting the $100 bill if you stop at a random location along the trail and pick up a bill .

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

  34. Virginia State Lottery

More Related