Stats SB Notes 42 Completednotebook February 22, 2017 Chapter 4 Discrete Probability Distributions Chapter Outline 41 Probability Distributions 42 Binomial Distributions 43 More Discrete Probability Distributions
Stats SB Notes 42 Completednotebook February 22, 2017 Section 42 Binomial Distributions Section 42 Objectives 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
Stats SB Notes 42 Completednotebook February 22, 2017 Binomial Experiments The experiment is repeated for a fixed number of trials, where each trial is independent of other trials 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) The probability of a success, P(S), is the same for each trial The random variable x counts the number of successful trials Notation for Binomial Experiments 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
Stats SB Notes 42 Completednotebook February 22, 2017 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 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 Solution: Binomial Experiments Binomial Experiment Each surgery represents a trial There are eight surgeries, and each one is independent of the others There are only two possible outcomes of interest for each surgery: a success (S) or a failure (F) The probability of a success, P(S), is 085 for each surgery The random variable x counts the number of successful surgeries
Stats SB Notes 42 Completednotebook February 22, 2017 Solution: 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, 4, 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 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
Stats SB Notes 42 Completednotebook February 22, 2017 Solution: 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 Try It Yourself 1, pg 202 Is the experiment binomial? n = p = q = x = Feb 22 12:35 PM
Stats SB Notes 42 Completednotebook February 22, 2017 Binomial Probability Formula Binomial Probability Formula The probability of exactly x successes in n trials is 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 Example: Finding Binomial Probabilities 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
Stats SB Notes 42 Completednotebook February 22, 2017 Expl 2 - Long Method Feb 22 12:38 PM Solution: Finding Binomial Probabilities Method 1: Draw a tree diagram and use the Multiplication Rule
Stats SB Notes 42 Completednotebook February 22, 2017 Solution: Finding Binomial Probabilities Method 2: Binomial Probability Formula Try It Yourself 2, pg 203 A card is selected from a standard deck and replaced This experiment is repeated a total of five times Find the probability of selecting exactly three clubs Feb 22 12:40 PM
Stats SB Notes 42 Completednotebook February 22, 2017 Binomial Probability Distribution Binomial Probability Distribution 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 0141 0422 0422 > Use binomial probability formula to find probabilities Example: Constructing a Binomial Distribution 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
Stats SB Notes 42 Completednotebook February 22, 2017 Solution: Constructing a Binomial Distribution 56% of adults like texting because it is quicker than calling n = 7, p = 056, q = 044, x = 0, 1, 2, 3, 4, 5, 6, 7 P(x = 0) = 7 C 0 (056) 0 (044) 7 = 1(056) 0 (044) 7 00032 P(x = 1) = 7 C 1 (056) 1 (044) 6 = 7(056) 1 (044) 6 00284 P(x = 2) = 7 C 2 (056) 2 (044) 5 = 21(056) 2 (044) 5 01086 P(x = 3) = 7 C 3 (056) 3 (044) 4 = 35(056) 3 (044) 4 02304 P(x = 4) = 7 C 4 (056) 4 (044) 3 = 35(056) 4 (044) 3 02932 P(x = 5) = 7 C 5 (056) 5 (044) 2 = 21(056) 5 (044) 2 02239 P(x = 6) = 7 C 6 (056) 6 (044) 1 = 7(056) 6 (044) 1 00950 P(x = 7) = 7 C 7 (056) 7 (044) 0 = 1(056) 7 (044) 0 00173 x Solution: Constructing a Binomial Distribution All of the probabilities are between 0 and 1 and the sum of the probabilities is 100001 1 P(x) 0 00032 1 00284 2 01086 3 02304 4 02932 5 02239 6 00950 7 00173
Stats SB Notes 42 Completednotebook February 22, 2017 Try It Yourself 3 Seven adults who participated in the survey are randomly selected and asked whether they use a tablet to access social media Construct a binomial distribution for the number of adults who respond yes Feb 22 12:47 PM Example: Finding Binomial Probabilities Using Technology 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) Solution: Binomial with n = 100, p = 057, x = 75
Stats SB Notes 42 Completednotebook February 22, 2017 Solution: Finding Binomial Probabilities Using Technology From the displays, you can see that the probability that exactly 75 adults consider air conditioning a necessity is about 002 Try It Yourself 4, pg 205 A survey found that 34% of US adults have hidden purchases from their spouses You randomly select 200 adults with spouses What is the probability that exactly 68 of them have hidden purchases from their spouses? Use technology to find the probability Feb 22 12:49 PM
Stats SB Notes 42 Completednotebook February 22, 2017 Example: Finding Binomial Probabilities Using Formulas A survey indicates that 41% 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 Solution: Finding Binomial Probabilities Using Formulas P(x = 2) = 4 C 2 (041) 2 (059) 2 = 6(041) 2 (059) 2 0351094 P(x = 3) = 4 C 3 (041) 3 (059) 1 = 4(041) 3 (059) 1 0162654 P(x = 4) = 4 C 4 (041) 4 (059) 0 = 1(041) 4 (059) 0 0028258 P(x 2) = P(2) + P(3) + P(4) 0542 0351094 + 0162654 + 0028258
Stats SB Notes 42 Completednotebook February 22, 2017 Try It Yourself 5, pg 206 Feb 22 2:20 PM Example: Finding Binomial Probabilities 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) Solution: Binomial with n = 8, p = 010, x = 4
Stats SB Notes 42 Completednotebook February 22, 2017 Solution: Finding Binomial Probabilities Using a Table A portion of Table 2 is shown The probability that exactly four of the eight workers carpool to work is 0005 Try It Yourself 6, pg 207 About 55% of all small businesses in the United States have a website You randomly select 10 small businesses What is the probability that exactly four of them have a website? Use a table to find the probability Feb 22 2:21 PM
Stats SB Notes 42 Completednotebook February 22, 2017 Example: Graphing a Binomial Distribution 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) Solution: Graphing a Binomial Distribution Histogram: x 0 1 2 3 4 5 6 P(x) 0004 0037 0138 0276 0311 0187 0047
Stats SB Notes 42 Completednotebook February 22, 2017 Try It Yourself 7, pg 208 A recent study found that 19% of people (ages 16 and older) in the United States own an e-reader You randomly select four people (ages 16 and older) and ask them whether they own an e- reader Construct a probability distribution for the random variable x Then graph the distribution Feb 22 2:24 PM Mean, Variance, and Standard Deviation Mean: μ = np Variance: σ 2 = npq Standard Deviation:
Stats SB Notes 42 Completednotebook February 22, 2017 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) Solution: Finding the Mean, Variance, and Standard Deviation μ = 168 σ 2 74 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) =114; 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
Stats SB Notes 42 Completednotebook February 22, 2017 Try It Yourself 8, pg 209 In San Francisco, California, about 44% of the days in a year are clear Find the mean, variance, and standard deviation for the number of clear days during the month of May Interpret the results and determine any unusual events Feb 22 2:27 PM Section 42 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
Stats SB Notes 42 Completednotebook February 22, 2017 Stats HW Section 42 pg 210, 1-10, 12-32 Evens Feb 22 2:29 PM