1. 6- 1 Chapter Six McGraw-Hill/Irwin © 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.

2. 6- 2 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.

3. 6- 3 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.

4. 6- 4 Types of Probability Distributions Discrete probability Distribution Discrete probability Distribution Can assume only certain outcomes Continuous Probability Distribution Continuous Probability Distribution Can assume an infinite number of values within a given range Types of Probability Distributions Probability Distribution A listing of all possible outcomes of an experiment and the corresponding probability. Random variable A numerical value determined by the outcome of an experiment.

5. 6- 5 Movie Continuous Probability Distribution

6. 6- 6 Features of a Discrete Distribution Discrete Probability Distribution Discrete Probability Distribution The sum of the probabilities of the various outcomes is 1.00. The probability of a particular outcome is between 0 and 1.00. The outcomes are mutually exclusive. The number of students in a class The number of children in a family The number of cars entering a carwash in a hour

7. 6- 7 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. TTT, TTH, THT, THH, HTT, HTH, HHT, HHH Example 1 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.

8. 6- 8 EXAMPLE 1 continued The outcome of zero heads occurred once. The outcome of one head occurred three times. The outcome of two heads occurred three times. The outcome of three heads occurred once.

9. 6- 9 The Mean of a Discrete Probability Distribution Mean Mean The central location of the data The long-run average value of the random variable Also referred to as its expected value, E(X), in a probability distribution A weighted average where  represents the mean  P(x) is the probability of the various outcomes x.

10. 6- 10 The Variance of a Discrete Probability Distribution Variance Measures the amount of spread (variation) of a distribution Denoted by the Greek letter s 2 (sigma squared) Standard deviation is the square root of s 2.

11. 6- 11 # houses Painted # of weeks Percent of weeks 10520 (5/20) 11630 (6/20) 12735 (7/20) 13210 (2/20) Total percent100 (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.

12. 6- 12 EXAMPLE 2 # houses painted (x) Probability P(x)x*P(x) 10.252.5 11.303.3 12.354.2 13.101.3  11.3 Mean number of houses painted per week

13. 6- 13 # houses painted (x) Probability P(x) (x-  (x-   (x-   P(x) 10.2510-11.31.69.423 11.3011-11.3.09.027 12.3512-11.3.49.171 13.1013-11.32.89.289   .910 Variance in the number of houses painted per week

14. 6- 14 Binomial Probability Distribution The trials are independent. 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.

15. 6- 15 Binomial Probability Distribution n! x!(n-x)! n is the number of trials x is the number of observed successes p is the probability of success on each trial Binomial Probability Distribution

16. 6- 16 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?

17. 6- 17 Example 3 The probability at least one is unemployed?

18. 6- 18 Mean & Variance of the Binomial Distribution Mean of the Binomial Distribution Variance of the Binomial Distribution

19. 6- 19 Example 3 Revisited Mean and Variance Example Recall that  =.2 and n=14  = n  = 14(.2) = 2.8  2 = n  (1-  ) = (14)(.2)(.8) =2.24

20. 6- 20 Finite Population The number of homes built in Blackmoor A population consisting of a fixed number of known individuals, objects, or measurements Finite Population The number of students in this class The number of cars in the parking lot

21. 6- 21 Hypergeometric Distribution Only 2 possible outcomes Sampling from a finite population without replacement, the probability of a success is not the same on each trial. Results from a count of the number of successes in a fixed number of trials Trials are not independent

22. 6- 22 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

23. 6- 23 Hypergeometric Distribution The size of the sample n is greater than 5% of the size of the population N 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). Use to find the probability of a specified number of successes or failures if

24. 6- 24 EXAMPLE 5 What is the probability that three of five violations randomly selected to be investigated are actually violations? 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.

25. 6- 25 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

26. 6- 26 Poisson probability distribution 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 where n is the number of trials p the probability of a success Variance Also equal to np Poisson Probability Distribution  = np

27. 6- 27 EXAMPLE 6 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?