Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall

Chapter 13 Counting Methods and Probability Theory

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 13.5 Probability with the Fundamental Counting Principle, Permutations, and Combinations

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Five groups agree to randomly select the order of performance by picking cards out of a hat, one at a time. What is the probability of the Rolling Stones performing fourth and the Beatles last? Solution Example continued

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Use the Fundamental Counting Principle to find the total number of possible permutations. Use the Fundamental Counting Principle to find the number of permutations with the Rolling Stones performing fourth and the Beatles performing last. continued

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall There are 3·2·1·1·1 = 6 possible permutations. Putting both values in the original equation. P (Rolling Stones fourth, Beatles last) =

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall A lottery game is set up so that each player chooses five different numbers from 1 to 60. With one lottery ticket, what is the probability of winning this prize? Solution Because the order of the five numbers does not matter, this situation involves combinations. Example continued

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Using the combinations formula, r = 5 and n = 60. If a person buys only one ticket, then that person has selected only one combination, thus P ≈

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall A club consists of five men and seven women. Three members are selected at random to attend a conference. Find the probability that the selected group consists of 3 men. Solution Order of selection does not matter, so this is a problem involving combinations. Example continued

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Consider the denominator: We are selecting r = 3 people from a total group of n = 12. Consider the numerator. We are interested in selecting 3 (r = 3) men from 5 (n = 5) men. Therefore: P(3 men) =