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Probabilistic and Statistical Techniques Lecture (14) Eng. Ismail Zakaria El Daour 2012 1 Probabilistic and Statistical Techniques Chapter 4 (part 1) Probability Distribution 2 Probabilistic and Statistical Techniques Random Variables Discrete random variable Continuous random variable 3 Probabilistic and Statistical Techniques Requirements for Probability Distribution P( x ) 1 where x assumes all possible values 0 P(x) 1 for every individual value of x 4 Probabilistic and Statistical Techniques Chapter 4 (part 2) Probability Distribution 5 Probabilistic and Statistical Techniques Mean, Variance and Standard Deviation of a Probability Distribution x.P( x ) Mean 2 ( x )2 .P( x ) Variance 2 [ x2 .P( x)] 2 Variance (shortcut) x .P( x ) 2 2 Standard Deviation 6 Probabilistic and Statistical Techniques Chapter 4 (part 3) Probability Distribution 7 Probabilistic and Statistical Techniques Binomial Probability Distributions 8 Probabilistic and Statistical Techniques Key Concept This section presents a basic definition of a binomial distribution along with notation, and it presents methods for finding probability values. Binomial probability distributions allow us to deal with circumstances in which the outcomes belong to two relevant categories such as acceptable/defective or survived/died. 9 Probabilistic and Statistical Techniques Definitions A binomial probability distribution results from a procedure that meets all the following requirements: 1. The procedure has a fixed number of trials. 2. The trials must be independent. (The outcome of any individual trial doesn’t affect the probabilities in the other trials.) 3. Each trial must have all outcomes classified into two categories (commonly referred to as success and failure). 4. The probability of a success remains the same in all trials. 10 11 Probabilistic and Statistical Techniques Notation for Binomial Probability Distributions S and F (success and failure) denote two possible categories of all outcomes; p and q will denote the probabilities of S and F, respectively, so P(S) = p (p = probability of success) P(F) = 1 – p = q (q = probability of failure) 12 Probabilistic and Statistical Techniques Notation (cont) n denotes the number of fixed trials. x denotes a specific number of successes in n trials, so x can be any whole number between 0 and n, inclusive. p denotes the probability of success in one of the n trials. q denotes the probability of failure in one of the n trials. P(x) denotes the probability of getting exactly x successes among the n trials. 13 Probabilistic and Statistical Techniques Binomial distribution Mean, Variance & Standard deviation 14 Probabilistic and Statistical Techniques Methods for Finding Probabilities Using the Binomial Probability Formula n x P( x) n C x p q x x 0,1,2,...,n where n = number of trials x = number of successes among n trials p = probability of success in any one trial q = probability of failure in any one trial (q = 1 – p) 15 Probabilistic and Statistical Techniques Methods for Finding Probabilities Using the Binomial Probability Formula 16 Probabilistic and Statistical Techniques Example 1 Use the binomial probability formula to find the probability of getting exactly 3 correct responses among 5 different requests from directory assistance. Assume that in general the responses is correct 90% of the time. That is Find P(3) given that n=5, x=3, p=0.9 & q=0.1 n x P( x) C p q n x x P(3)5 C3 P 3 q 53 P(3)5 C3 0.9 0.1 53 10 0.729 0.01 0.0729 3 17 Probabilistic and Statistical Techniques Example 2 Consider the experiment of flipping a coin 3 times. If we let the event of getting tails on a flip be considered “success”, and if the random variable T represents the number of tails obtained, then T will be binomially distributed with n=3 ,p=0.5 , and q=0.5 . calculate the probability of exactly 2 tails 2 1 1 1 3 1 3 P(2) 3 C2 3 .375 2 2 2 8 18 Probabilistic and Statistical Techniques Example 3 Each sample of water has a 10% chance of containing a particular organic pollutant. Assume that the samples are independent with regard to the presence of the pollutant. Find the probability that in the next 18 samples, exactly 2 contain the pollutant. Let n=18, x=2, p=0.1 & q=0.9 n x P( x ) n C x P q x P(3)18 C2 P 2 q16 P(3)18 C2 0.1 0.9 2 16 19 Probabilistic and Statistical Techniques Example 3 (cont.) Also, Find the probability that in the next 18 samples, that 3 or 4 contain the pollutant. Let n=18, p=0.1 & q=0.9 P(3) P(4) 18 C3 P3 q (183) 18 C4 P 4 q (184) Find mean & standard deviation np 18(0.1) 1.8 np(1 p) 1.8(1 0.1) 1.27 20 Probabilistic and Statistical Techniques Using table to Calculate the Binomial Probability Lecture 16 21 Probabilistic and Statistical Techniques Example 4 There is a 0.54 probability that a randomly selected freshman at a two-year college will return the second year. In each case, assume that 5 freshmen at a two-year college are randomly selected and find the probability indicated • Find the probability that at least four of the freshmen return for the second year • Find the probability that at most two of the freshmen return for the second year • Find the probability that more than one of the freshmen return for the second year 22 Probabilistic and Statistical Techniques Solution: P(4) P(5) 5 C4 0.54 0.46 5 C5 0.54 0.46 4 1 5 0 P(0) P(1) P(2) P(2) P(3) P(4) P(5) 23 Probabilistic and Statistical Techniques Example 5 The rate of on-time flights for commercial jets are continuously tracked by the U.S. Department of Transportation. Recently, Southwest Air had the best rate with 80% of its flights arriving on time. A test is conducted by randomly selecting 15 southwest flights and observing whether they arrive on time. • Find the probability that exactly 10 flights arrive on time • Find the probability that at least 10 flights arrive late 24 Probabilistic and Statistical Techniques Solution: P (10 )15 C10 0.810 0.25 P(10) P(11) P(12) P(13) P(14) P(15) Where success = arrive late OR P(0) P(1) P(2) P(3) P(4) P(5) Where success = arrive on time 25 Probabilistic and Statistical Techniques Example 6 Nine Percent of men and 0.25% of women cannot distinguish between the colors red and green. This is a type of color blindness that cause problems with traffic signals. If 6 men are randomly selected for a study of traffic signals perceptions, find the probability that exactly two of them cannot distinguish between red and green. 26 Probabilistic and Statistical Techniques Solution: P (2) 6 C2 0.09 2 * 0.914 P(2) 0.0833 27 Probabilistic and Statistical Techniques Example 7 In a housing study, it was found that 26% of college students live in campus housing. The providence Insurance Company wants to sell those students special policies insuring their personal property. If they test a marketing strategy by randomly selecting six college students, • what is the probability that at least one of them live in campus housing • what is the mean of no. of students live in campus housing 28 Probabilistic and Statistical Techniques Example 8 Air America has a policy of booking as many as 15 persons on an airplane that can seat only 14. Because past studies have revealed that only 85% of the booked passengers actually arrive for the flight. Find the Probability that if Air America books 15 persons, not enough seats will be available. 29 Probabilistic and Statistical Techniques Example 9 An automobile manufacturer has determined that 30% of all gas tanks that were installed on its 2002 compact model are defective. If 14 of these cars are independently sampled, what is the probability that more than 11 cars need new gas tanks? 30 Probabilistic and Statistical Techniques Example 10 There are five flights daily from Pittsburgh via US airways into the Bradford, Pennsylvania Regional Airport. Suppose the probability that any flight arrives late is 0.2. what is the probability that none of the flights are late today? What is the probability that exactly one of the flights is late today? 31 Probabilistic and Statistical Techniques Example 11 Suppose 60 percent of people prefer Coke to Pepsi. We select 18 people for further study. • How many would you expect to prefer Coke ? • What is the probability 10 of those surveyed will prefer Coke ? • What is the probability 15 prefer Coke ? • Compute the mean and the standard deviation 32 Probabilistic and Statistical Techniques Example 12 In a recent study 90% of the homes in the United States were found to have large screen TVs. In a sample of nine homes, what is the probability that: • All nine have large screen TVs • Less than five have large screen TVs • More than five have large screen TVs • at least seven homes have large screen TVs 33 Probabilistic and Statistical Techniques Example 13 A company receive 60% of its order over the internet. Within a collection of 18 independently placed orders. What is the probability that: • Between eight and ten (inclusive) of the orders are received over the internet • no more than four of the orders are received over the internet 34 35

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