The Equally Likely Approach and Associated Counting Methods

The Equally Likely Approach and Associated Counting Methods – Combinations and Permutations

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

Dice You toss two dice. What is the probability of the outcomes summing to 5? This is S: {(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

We Need Efficient Methods for Counting Outcomes

NCAA Tournament “Bracket”

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: 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

Counting Example z. Pollsters minimize lead-in effect by rearranging the order of the questions on a survey z. 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?

Solution z. 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. z. The number of possible arrangements is therefore 5 4 3 2 1 = 120

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

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

Factorial Examples z 20! = 2. 43 x 1018 z 1, 000 seconds? z. About 11. 5 days z 1, 000, 000 seconds? z. About 31 years z 31 years = 109 seconds z 1018 = 109 x 109 z 20! is roughly the age (according to some) of the universe in seconds

Permutations A B C D E z. 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? z 5 4 = 20

Permutations (cont. )

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

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

Combinations (cont. )

Example: Illinois State Lottery

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

The Forrest Gump Visualization of Your Lottery Chances z. How large is 195, 249, 054? z$1 bill and $100 bill both 6” in length z 10, 560 bills = 1 mile z. Let’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

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

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

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

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

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

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

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

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

Virginia State Lottery