Chapter Discrete Probability s Chapter Outline 1 Probability s 2 Binomial s 3 More Discrete Probability s Copyright 2015, 2012, and 2009 Pearson Education, Inc 1 Copyright 2015, 2012, and 2009 Pearson Education, Inc 2 Section 2 Objectives Section 2 Binomial s How to determine whether a probability experiment is a binomial experiment How to find binomial probabilities using the binomial probability formula How to find binomial probabilities using technology, formulas, and a binomial table How to construct and graph a binomial distribution How to find the mean, variance, and standard deviation of a binomial probability distribution Copyright 2015, 2012, and 2009 Pearson Education, Inc 3 Copyright 2015, 2012, and 2009 Pearson Education, Inc Binomial Experiments Notation for Binomial Experiments 1 The experiment is repeated for a fixed number of trials, where each trial is independent of other trials 2 There are only two possible outcomes of interest for each trial The outcomes can be classified as a success (S) or as a failure (F) 3 The probability of a success, P(S), is the same for each trial The random variable x counts the number of successful trials Symbol n p = P(S) q = P(F) x Description The number of times a trial is repeated The probability of success in a single trial The probability of failure in a single trial (q = 1 p) The random variable represents a count of the number of successes in n trials: x = 0, 1, 2, 3,, n Copyright 2015, 2012, and 2009 Pearson Education, Inc 5 Copyright 2015, 2012, and 2009 Pearson Education, Inc 6 Larson/Farber 5 th ed 1
Example: Binomial Experiments Decide whether the experiment is a binomial experiment If it is, specify the values of n, p, and q, and list the possible values of the random variable x 1 A certain surgical procedure has an 85% chance of success A doctor performs the procedure on eight patients The random variable represents the number of successful surgeries Binomial Experiments Binomial Experiment 1 Each surgery represents a trial There are eight surgeries, and each one is independent of the others 2 There are only two possible outcomes of interest for each surgery: a success (S) or a failure (F) 3 The probability of a success, P(S), is 085 for each surgery The random variable x counts the number of successful surgeries Copyright 2015, 2012, and 2009 Pearson Education, Inc 7 Copyright 2015, 2012, and 2009 Pearson Education, Inc 8 Binomial Experiments Binomial Experiment n = 8 (number of trials) p = 085 (probability of success) q = 1 p = 1 085 = 015 (probability of failure) x = 0, 1, 2, 3,, 5, 6, 7, 8 (number of successful surgeries) Example: Binomial Experiments Decide whether the experiment is a binomial experiment If it is, specify the values of n, p, and q, and list the possible values of the random variable x 2 A jar contains five red marbles, nine blue marbles, and six green marbles You randomly select three marbles from the jar, without replacement The random variable represents the number of red marbles Copyright 2015, 2012, and 2009 Pearson Education, Inc 9 Copyright 2015, 2012, and 2009 Pearson Education, Inc 10 Binomial Experiments Not a Binomial Experiment The probability of selecting a red marble on the first trial is 5/20 Because the marble is not replaced, the probability of success (red) for subsequent trials is no longer 5/20 The trials are not independent and the probability of a success is not the same for each trial Binomial Probability Formula Binomial Probability Formula The probability of exactly x successes in n trials is n! P( x) = ncx p q = p q ( n x)! x! n = number of trials p = probability of success q = 1 p probability of failure x = number of successes in n trials Note: number of failures is n x x n x x n x Copyright 2015, 2012, and 2009 Pearson Education, Inc 11 Copyright 2015, 2012, and 2009 Pearson Education, Inc 12 Larson/Farber 5 th ed 2
Microfracture knee surgery has a 75% chance of success on patients with degenerative knees The surgery is performed on three patients Find the probability of the surgery being successful on exactly two patients Method 1: Draw a tree diagram and use the Multiplication Rule 9 P(2 successful surgeries) = 3 022 6 Copyright 2015, 2012, and 2009 Pearson Education, Inc 13 Copyright 2015, 2012, and 2009 Pearson Education, Inc 1 Method 2: Binomial Probability Formula 3 1 n = 3, p =, q = 1 p =, x = 2 2 3 2 3 1 = 3 2 P(2 successful surgeries) C 2 1 3! 3 1 = (3 2)!2! 9 1 27 = 3 = 022 16 6 Binomial Probability Binomial Probability List the possible values of x with the corresponding probability of each Example: Binomial probability distribution for Microfacture knee surgery: n = 3, p = x 0 1 2 3 P(x) 0016 011 022 022 Use binomial probability formula to find probabilities 3 Copyright 2015, 2012, and 2009 Pearson Education, Inc 15 Copyright 2015, 2012, and 2009 Pearson Education, Inc 16 Example: Constructing a Binomial In a survey, US adults were asked to give reasons why they liked texting on their cellular phones Seven adults who participated in the survey are randomly selected and asked whether they like texting because it is quicker than Calling Create a binomial probability distribution for the number of adults who respond yes Constructing a Binomial 56% of adults like texting because it is quicker than calling n = 7, p = 056, q = 0, x = 0, 1, 2, 3,, 5, 6, 7 P(x = 0) = 7 C 0 (056) 0 (0) 7 = 1(056) 0 (0) 7 00032 P(x = 1) = 7 C 1 (056) 1 (0) 6 = 7(056) 1 (0) 6 0028 P(x = 2) = 7 C 2 (056) 2 (0) 5 = 21(056) 2 (0) 5 01086 P(x = 3) = 7 C 3 (056) 3 (0) = 35(056) 3 (0) 0230 P(x = ) = 7 C (056) (0) 3 = 35(056) (0) 3 02932 P(x = 5) = 7 C 5 (056) 5 (0) 2 = 21(056) 5 (0) 2 02239 P(x = 6) = 7 C 6 (056) 6 (0) 1 = 7(056) 6 (0) 1 00950 P(x = 7) = 7 C 7 (056) 7 (0) 0 = 1(056) 7 (0) 0 00173 Copyright 2015, 2012, and 2009 Pearson Education, Inc 17 Copyright 2015, 2012, and 2009 Pearson Education, Inc 18 Larson/Farber 5 th ed 3
Constructing a Binomial x P(x) 0 00032 1 0028 2 01086 3 0230 02932 5 02239 6 00950 7 00173 All of the probabilities are between 0 and 1 and the sum of the probabilities is 100001 1 The results of a recent survey indicate that 67% of US adults consider air conditioning a necessity If you randomly select 100 adults, what is the probability that exactly 75 adults consider air conditioning a necessity? Use a technology tool to find the probability (Source: Opinion Research Corporation) Binomial with n = 100, p = 057, x = 75 Copyright 2015, 2012, and 2009 Pearson Education, Inc 19 Copyright 2015, 2012, and 2009 Pearson Education, Inc 20 Using Formulas A survey indicates that 1% of women in the US consider reading their favorite leisure-time activity You randomly select four US women and ask them if reading is their favorite leisure-time activity Find the probability that at least two of them respond yes From the displays, you can see that the probability that exactly 75 adults consider air conditioning a necessity is about 002 n =, p = 01, q = 059 At least two means two or more Find the sum of P(2), P(3), and P() Copyright 2015, 2012, and 2009 Pearson Education, Inc 21 Copyright 2015, 2012, and 2009 Pearson Education, Inc 22 Using Formulas P(x = 2) = C 2 (01) 2 (059) 2 = 6(01) 2 (059) 2 035109 P(x = 3) = C 3 (01) 3 (059) 1 = (01) 3 (059) 1 016265 P(x = ) = C (01) (059) 0 = 1(01) (059) 0 0028258 P(x 2) = P(2) + P(3) + P() 035109 + 016265 + 0028258 052 The results of a recent survey indicate that 67% of US adults consider air conditioning a necessity If you randomly select 100 adults, what is the probability that exactly 75 adults consider air conditioning a necessity? Use a technology tool to find the probability (Source: Opinion Research Corporation) Binomial with n = 100, p = 057, x = 75 Copyright 2015, 2012, and 2009 Pearson Education, Inc 23 Copyright 2015, 2012, and 2009 Pearson Education, Inc 2 Larson/Farber 5 th ed
Using a Table About ten percent of workers (16 years and over) in the United States commute to their jobs by carpooling You randomly select eight workers What is the probability that exactly four of them carpool to work? Use a table to find the probability (Source: American Community Survey) From the displays, you can see that the probability that exactly 75 adults consider air conditioning a necessity is about 002 Binomial with n = 8, p = 010, x = Copyright 2015, 2012, and 2009 Pearson Education, Inc 25 Copyright 2015, 2012, and 2009 Pearson Education, Inc 26 Using a Table A portion of Table 2 is shown The probability that exactly four of the eight workers carpool to work is 0005 Example: Graphing a Binomial Sixty percent of households in the US own a video game console You randomly select six households and ask each if they own a video game console Construct a probability distribution for the random variable x Then graph the distribution (Source: Deloitte, LLP) n = 6, p = 06, q = 0 Find the probability for each value of x Copyright 2015, 2012, and 2009 Pearson Education, Inc 27 Copyright 2015, 2012, and 2009 Pearson Education, Inc 28 Graphing a Binomial x 0 1 2 3 5 6 P(x) 000 0037 0138 0276 0311 0187 007 Histogram: Mean, Variance, and Standard Deviation Mean: μ = np Variance: σ 2 = npq Standard Deviation: σ = npq Copyright 2015, 2012, and 2009 Pearson Education, Inc 29 Copyright 2015, 2012, and 2009 Pearson Education, Inc 30 Larson/Farber 5 th ed 5
Example: Finding the Mean, Variance, and Standard Deviation In Pittsburgh, Pennsylvania, about 56% of the days in a year are cloudy Find the mean, variance, and standard deviation for the number of cloudy days during the month of June Interpret the results and determine any unusual values (Source: National Climatic Data Center) n = 30, p = 056, q = 0 Mean: μ = np = 30 056 = 168 Variance: σ 2 = npq = 30 056 0 7 Standard Deviation: σ = npq = 30 056 0 27 Finding the Mean, Variance, and Standard Deviation μ = 168 σ 2 7 σ 27 On average, there are 168 cloudy days during the month of June The standard deviation is about 27 days Values that are more than two standard deviations from the mean are considered unusual 168 2(27) =11; A June with 11 cloudy days or less would be unusual 168 + 2(27) = 222; A June with 23 cloudy days or more would also be unusual Copyright 2015, 2012, and 2009 Pearson Education, Inc 31 Copyright 2015, 2012, and 2009 Pearson Education, Inc 32 Section 2 Summary Determined if a probability experiment is a binomial experiment Found binomial probabilities using the binomial probability formula Found binomial probabilities using technology, formulas, and a binomial table Constructed and graphed a binomial distribution Found the mean, variance, and standard deviation of a binomial probability distribution Copyright 2015, 2012, and 2009 Pearson Education, Inc 33 Larson/Farber 5 th ed 6