4.2 Probability Models

4.1 (cont.) Probability ModelsThe Equally Likely Approach(also called the Classical Approach)Assigning ProbabilitiesIf an experiment has N outcomes, then each outcome has probability 1/N of occurringIf an event A1 has n1 outcomes, thenP(A1) = n1/N

DiceYou 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 = 0.111This is S:

{(1,1), (1,2), (1,3), etc.}

We Need Efficient Methods for Counting OutcomesProduct Rule for Ordered PairsA 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?Product Rule for Ordered Pairsjunior colleges: 1, 2, 3, 4state colleges a, b, cpossible 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)Product Rule for Ordered Pairsjunior colleges: 1, 2, 3, 4state colleges a, b, cpossible 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 colleges3 state collegestotal number of possiblepairs = 4 x 3 = 12Product Rule for Ordered Pairsjunior colleges: 1, 2, 3, 4state colleges a, b, cpossible 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 waysto choose the first element ofthe pair, and n2 ways to choosethe second element, then the number of possible pairs isn1n2. Here n1 = 4, n2 = 3.Counting in Either-Or SituationsNCAA Basketball Tournament, 68 teams: how many ways can the bracket be filled out?How many games?2 choices for each gameNumber of ways to fill out the bracket:267 = 1.5 1020Earth pop. about 6 billion; everyone fills out 100 million different bracketsChances of getting all games correct is about 1 in 1,000

A states 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?

1,3006,5522,600,000786,24026,00010Counting ExamplePollsters minimize lead-in effect by rearranging the order of the questions on a surveyIf Gallup has a 5-question survey, how many different versions of the survey are required if all possible arrangements of the questions are included?SolutionThere 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 therefore5 4 3 2 1 = 120

Efficient Methods for Counting OutcomesFactorial Notation:n!=12 nExamples1!=1; 2!=12=2; 3!= 123=6; 4!=24;5!=120;Special definition: 0!=1

Factorials with calculators and ExcelCalculator: non-graphing: x ! (second function)graphing: bottom p. 9 T I Calculator Commands(math button)Excel:Insert function: Math and Trig category, FACT functionFactorial Examples20! = 2.43 x 10181,000,000 seconds?About 11.5 days1,000,000,000 seconds?About 31 years31 years = 109 seconds1018 = 109 x 10920! is roughly the age of the universe in secondsPermutationsA B C D EHow 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 = 20Permutations (cont.)

Permutations with calculator and ExcelCalculatornon-graphing: nPrGraphingp. 9 of T I Calculator Commands(math button)ExcelInsert function: Statistical, PermutCombinationsA B C D EHow 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 importantDivide by 2: (5 4)/2 = 10 waysCombinations (cont.)

BUS/ST 350 Powerball Lottery

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

Chances of Winning?

Example: Illinois State Lottery

North Carolina Powerball LotteryPrior to Jan. 1, 2009After Jan. 1, 2009

Most recent change: powerball number is from 1 to 35http://www.nc-educationlottery.org/faq_powerball.aspx#43The Forrest Gump Visualization of Your Lottery ChancesHow large is 195,249,054?$1 bill and $100 bill both 6 in length

10,560 bills = 1 mileLets 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

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

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 New York back to Raleigh

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

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 .More ChangesAfter Jan. 1, 2012After Oct. 4, 2015http://www.nc-educationlottery.org/powerball_how-to-play.aspx

Virginia State Lottery