Chapter Six McGraw-Hill/Irwin © 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.

Discrete Probability Distributions Chapter Six Discrete Probability Distributions GOALS 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.

Discrete Probability Distributions Chapter Six continued Discrete Probability Distributions GOALS 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.

Types of Probability Distributions A listing of all possible outcomes of an experiment and the corresponding probability. Discrete probability Distribution Can assume only certain outcomes Continuous Probability Distribution Can assume an infinite number of values within a given range Random variable A numerical value determined by the outcome of an experiment. Types of Probability Distributions

Continuous Probability Distribution Movie

Features of a Discrete Distribution Discrete Probability Distribution The sum of the probabilities of the various outcomes is 1.00. The outcomes are mutually exclusive. The probability of a particular outcome is between 0 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

The possible outcomes for such an experiment 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. 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. From the definition of a random variable, x as defined in this experiment, is a random variable. Example 1

The outcome of two heads occurred three times. The outcome of zero heads occurred once. The outcome of one head occurred three times. The outcome of three heads occurred once. EXAMPLE 1 continued

The Mean of a Discrete Probability Distribution The long-run average value of the random variable The central location of the data Also referred to as its expected value, E(X), in a probability distribution A weighted average where m represents the mean mP(x) is the probability of the various outcomes x. The Mean of a Discrete Probability Distribution

The Variance of a Discrete Probability Distribution Denoted by the Greek letter s2 (sigma squared) Measures the amount of spread (variation) of a distribution Standard deviation is the square root of s2. The Variance of a Discrete Probability Distribution

# houses Painted # of weeks Percent of weeks 10 5 20 (5/20) 11 6 30 (6/20) 12 7 35 (7/20) 13 2 10 (2/20) Total percent 100 (20/20) Dan Desch, owner of College Painters, studied his records for the past 20 weeks and reports the following number of houses painted per week.

Mean number of houses painted per week # houses painted (x) Probability P(x) x*P(x) 10 .25 2.5 11 .30 3.3 12 .35 4.2 13 .10 1.3 m = 11.3 Mean number of houses painted per week EXAMPLE 2

Variance in the number of houses painted per week # houses painted (x) Probability P(x) (x-m) (x-m)2 (x-m)2 P(x) 10 .25 10-11.3 1.69 .423 11 .30 11-11.3 .09 .027 12 .35 12-11.3 .49 .171 13 .10 13-11.3 2.89 .289 s2 = .910

Binomial Probability Distribution An outcome of an experiment is classified into one of two mutually exclusive categories, such as a success or failure. The probability of success stays the same for each trial. The data collected are the results of counts. The trials are independent. Binomial Probability Distribution

Binomial Probability Distribution n is the number of trials x is the number of observed successes p is the probability of success on each trial n! x!(n-x)! Binomial Probability Distribution

What is the probability that exactly three are unemployed? The Alabama Department of Labor reports that 20% of the workforce in Mobile is unemployed and interviewed 14 workers. What is the probability that exactly three are unemployed? At least three are unemployed?

The probability at least one is unemployed? Example 3

Mean & Variance of the Binomial Distribution Mean of the Binomial Distribution Variance of the Binomial Distribution Mean & Variance of the Binomial Distribution

Mean and Variance Example Example 3 Revisited Recall that p=.2 and n=14 m= np = 14(.2) = 2.8 s2 = n p(1- p ) = (14)(.2)(.8) =2.24 Mean and Variance Example

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

Hypergeometric Distribution Only 2 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

Hypergeometric Distribution 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

Hypergeometric Distribution 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

The Transportation Security Agency has a list of 10 reported safety violations. Suppose only 4 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 5

Poisson probability distribution The limiting form of the binomial distribution where the probability of success p 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

Poisson probability distribution m = np where n is the number of trials p the probability of a success where m 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

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 2 arrivals in an hour? EXAMPLE 6