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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 A1 has n1 outcomes, then P(A1) = n1/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 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? 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) Product Rule for Ordered Pairs junior colleges: 1, 2, 3, 4 state colleges a, b, c 4 junior colleges 3 state colleges possible pairs: total number of possible (1, a) (1, b) (1, c) pairs = 4 x 3 = 12 (2, a) (2, b) (2, c) (3, a) (3, b) (3, c) (4, a) (4, b) (4, c) Product Rule for Ordered Pairs junior colleges: 1, In 2,general, 3, 4 if there are n1 ways to choose the first element of state colleges a, b,thec pair, and n ways to choose 2 the second element, then the possible pairs: number of possible pairs is (1, a) (1, b) (1, c) n1n2. Here n1 = 4, n2 = 3. (2, a) (2, b) (2, c) (3, a) (3, b) (3, c) (4, a) (4, b) (4, c) 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 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 5digit number. How many different license plates can the state issue? 1. 2. 3. 4. 5. 1,300 6,552 2,600,000 786,240 26,000 0% 1 0% 0% 2 3 0% 0% 4 5 10 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? 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 Efficient Methods for Counting Outcomes Factorial Notation: n!=12 … n Examples 1!=1; 2!=12=2; 3!= 123=6; 4!=24; 5!=120; Special definition: 0!=1 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 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 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 Permutations (cont.) 5! 5! 5 4 20 5 4 (5 2)! 3! 5! Notation: 5 P2 20 (5 2)! Permutations with calculator and Excel Calculator non-graphing: nPr Graphing p. 9 of T I Calculator Commands (math button) Excel Insert function: Statistical, Permut 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 Combinations (cont.) 5! 5! 5 4 20 5 C2 10 (5 2)!2! 3!2! 1 2 2 n! n Cr (n r )!r! 5 2 n r 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? Choose6 numbersfrom 20, without replacement, order not important. Number of possibilities? 20 6 20! 20 C6 38,760 (20 6)!6! Example: Illinois State Lottery Choose6 numbersfrom54 numbers wit hout replacement ; order not import ant 54! 25,827,165 54 C6 48!6! (about 1 secondin 10 mont hs) (1200ft 2 house,16.5million ping pongballs) North Carolina Powerball Lottery Prior to Jan. 1, 2009 After Jan. 1, 2009 5 from 1 - 55: 5 from 1 - 59: 55! 3, 478, 761 5!50! 59! 5, 006, 386 5!54! 1 from 1 - 42 (p'ball #): 1 from 1 - 39 (p'ball #): 42! 42 1!41! 39! 39 1!38! 3, 478, 761*42 5, 006, 386*39 146,107, 962 195, 249, 054 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 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 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? 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 Changes After Jan. 1, 2009 5 from 1 - 59: 59! 5, 006,386 5!54! 1 from 1 - 39 (p'ball #): 39! 39 1!38! 5, 006,386*39 195, 249, 054 After Jan. 1, 2012 http://www.nceducationlottery.org/pow erball_how-to-play.aspx Virginia State Lottery 50! P ick 5 : 50 C5 2,118,760 45!5! 2,118,760 25 C1 25! 2,118,760 52,969000 24!1!