6- Chapter Six McGraw- © 2005 The McGraw-Hill Companies, Inc., All 6- Chapter Six Discrete Probability GOALSDistributions When you have completed this chapter, you will be able to: ONE Define the terms random variable and probability distribution TWO Distinguish between a discrete and continuous probability distributions THREE Calculate the mean, variance, and standard deviation of a discrete probability distribution Chapter Six 6- continued Discrete Probability GOALS Distributions When you have completed this chapter, you will be able to: FOUR Describe the characteristics and compute probabilities using the binomial probability distribution FIVE Describe the characteristics and compute probabilities using the hypergeometric distribution SIX Describe the characteristics and compute the probabilities using the Poisson distribution Probability Distribution 6- Types of Probability Distributions A listing of all possible outcomes Discrete probability Distribution of an experiment Can assume only certain and the outcomes corresponding probability Continuous Probability Distribution Can assume an infinite number of Random variable values within a given range A numerical value determined by the outcome of an experiment Types of Probability Distributions 6- Continuous Probability Distribution Movie Discrete Probability Distribution The sum of the probabilities of the various outcomes is 1.00 The outcomes are mutually exclusive 6- The probability of a particular outcome is between and 1.00 The number of students in a class The number of cars entering a carwash in a hour The number of children in a family Features of a Discrete Distribution 6- Consider a random experiment in which a coin is tossed three times Let x be the number of heads Let H represent the outcome of a head and T the outcome of a tail From the definition of a random variable, x as defined in this experiment, is a random variable The possible outcomes for such an experiment TTT, TTH, THT, THH, HTT, HTH, HHT, HHH Thus the possible values of x (number of heads) are 0,1,2,3 Example The outcome of two heads occurred three times The outcome of three heads occurred once 6- The outcome of zero heads occurred once The outcome of one head occurred three times EXAMPLE continued 6- Mean The long-run average value of the random variable The central location of the data [ xP( x)] A weighted average Also referred to as its expected value, E(X), in a probability distribution where represents the mean P(x) is the probability of the various outcomes x The Mean of a Discrete Probability Distribution 6- 10 Variance Measures the amount of spread (variation) of a distribution Denoted by the Greek letter s2 (sigma squared) Standard deviation is the square root of s2 [( x ) P( x)] The Variance of a Discrete Probability Distribution 6- 13 Variance in the number of 2 [( x ) P ( x)] houses painted per week # houses Probability painted (x) P(x) (x- (x- (x P(x) 10 11 25 30 10-11.3 11-11.3 1.69 09 423 027 12 13 35 10 12-11.3 13-11.3 49 2.89 171 289 910 6- 14 Binomial Probability Distribution An outcome of an experiment is classified into one of two mutually exclusive categories, such as a success or failure The data collected are the results of counts The probability of success stays the same for each trial The trials are independent Binomial Probability Distribution 6- 15 Binomial Probability Distribution x P( x)n C x (1 ) n x n is the number of trials x is the number of observed successes p is the probability of success on each trial n Cx n! x!(n-x)! Binomial Probability Distribution The Alabama Department of Labor reports that 20% of the workforce in Mobile is unemployed and interviewed 14 workers 6- 16 What is the probability that exactly three are unemployed? P( x 3)14 C3 (.20) (.80)11 14 C14 (.20)14 (.80) .250 172 000 .551 At least three are unemployed? P (3)14 C (.20) (1 20)11 (364)(.0080)(.0859) .2501 6- 17 The probability at least one is unemployed? P( x 1) 1 P(0) 14 1 14 C0 (.20) (1 20) 1 044 .956 Example 6- 18 Mean of the Binomial Distribution n Variance of the Binomial Distribution n (1 ) Mean & Variance of the Binomial Distribution 6- 19 Example Revisited Recall that =.2 and n=14 = n = 14(.2) = 2.8 = n (1- ) = (14)(.2)(.8) =2.24 Mean and Variance Example 6- 20 Finite Population A population consisting of a fixed number of known individuals, objects, or measurements The number of students in this class The number of cars in the parking lot The number of homes built in Blackmoor Finite Population 6- 21 Hypergeometric Distribution Only possible outcomes Results from a count of the number of successes in a fixed number of trials Trials are not independent Sampling from a finite population without replacement, the probability of a success is not the same on each trial Hypergeometric Distribution 6- 22 Hypergeometric Distribution ( S Cx )( N S Cn x ) P( x ) N Cn where N is the size of the population S is the number of successes in the population x is the number of successes in a sample of n observations Hypergeometric Distribution 6- 23 Use to find the probability of a specified number of successes or failures if The sample is selected from a finite population without replacement (recall that a criteria for the binomial distribution is that the probability of success remains the same from trial to trial) The size of the sample n is greater than 5% of the size of the population N Hypergeometric Distribution 6- 24 The Transportation Security Agency has a list of 10 reported safety violations Suppose only of the reported violations are actual violations and the Security Agency will only be able to investigate five of the violations What is the probability that three of five violations randomly selected to be investigated are actually violations? EXAMPLE 6- 25 ( C )( 10 C 5 P (3) 10 C ( C )( C ) 4(15) .238 252 10 C The limiting form of the binomial distribution where the probability of success is small and n is large is called the Poisson probability distribution The binomial distribution becomes more skewed to the right (positive) as the probability of success become smaller Poisson probability distribution 6- 26 Poisson Probability Distribution = np x u e P( x ) x! where n is the number of trials p the probability of a success where is the mean number of successes in a particular interval of time e is the constant 2.71828 x is the number of successes Variance Also equal to np Poisson probability distribution 6- 27 The Sylvania Urgent Care facility specializes in caring for minor injuries, colds, and flu For the evening hours of 6-10 PM the mean number of arrivals is 4.0 per hour What is the probability of arrivals in an hour? x u e P( x) x! 4 e .1465 2! EXAMPLE ... students in a class The number of cars entering a carwash in a hour The number of children in a family Features of a Discrete Distribution 6- Consider a random experiment in which a coin is tossed... Example 6- 20 Finite Population A population consisting of a fixed number of known individuals, objects, or measurements The number of students in this class The number of cars in the parking lot The... facility specializes in caring for minor injuries, colds, and flu For the evening hours of 6-10 PM the mean number of arrivals is 4.0 per hour What is the probability of arrivals in an hour? x u