PADM 7060 Quantitative Methods for Public Administration Unit 3 Chapters 9-10 Fall 2004 Jerry Merwin

Meier & Brudney Part III: Probability  Chapter 7: Introduction to Probability  Chapter 8: The Normal Probability Distribution  Chapter 9: The Binomial Probability Distribution  Chapter 10: Some Special Probability Distributions

Meier & Brudney: Chapter 9 The Binomial Probability Distribution  What is the binomial probability distribution?  What is a Bernoulli process?

Meier & Brudney: Chapter 9 The Binomial Probability Distribution (Page 2)  Let’s discuss the characteristics of a Bernoulli process: (book says two, but can be three) Each trial’s outcome must be mutually exclusive: Success or failure Probability of success (p) remains constant (as does probability of failure, q) Trials are independent (Not affected by outcomes of earlier trials) Examples:  Coin flip  Roll of die (six or not six)  Fire in a community

Meier & Brudney: Chapter 9 The Binomial Probability Distribution (Page 3)  More on Bernoulli process - Must how three things to determine probability: Number of trials Number of successes Probability of success in any trial  Formula (see page 146) Combination of n things taken r at a time

Meier & Brudney: Chapter 9 The Binomial Probability Distribution (Page 4)  Let’s look at some examples: Starting at the bottom of page 146:  How many sets of three balls can be selected from four balls (a, b, c, d)?  How many combinations of four things two at a time?  We can list the possibilities with these examples to come up with solutions.

Meier & Brudney: Chapter 9 The Binomial Probability Distribution (Page 5)  What if we do not want to list all the combinations? See the formula on page 147  Do you understand how to calculate a factorial? Examples of formula with coin flip (pages )

Meier & Brudney: Chapter 9 The Binomial Probability Distribution (Page 6)  What is the problem with calculating factorials? Cumbersome in larger numbers!  So what alternative do we have?  How can the normal curve be used? A general rule ‑ of ‑ thumb is that once n ≥ 30, the normal distribution can be used to calculate the binomial probability distribution.

Meier & Brudney: Chapter 9 The Binomial Probability Distribution (Page 7)  How about the problem on the Republican’s chance of being hired in Chicago? (Page 151) See the information for the binomial distribution The standard deviation of a probability distribution at the bottom of 151 Simply convert the number actually hired into a z score and look up the value.

Meier & Brudney: Chapter 9 The Binomial Probability Distribution (Page 8)  On page 152, you're told that you should use this only when n x p is greater than 10 and n x (1 -p) is greater than 10.  Also, keep in mind that the normal curve tells you the probability only that r or fewer (or more) events occurred. It does not tell you the probability of exactly r events occurring.  This probability can be determined only by using the binomial distribution.

Meier & Brudney: Chapter 9 The Binomial Probability Distribution (Page 9)  Problems: 9.2 & 9.6

Meier & Brudney: Chapter 10 Special Probability Distributions  What is the Hypergeometric Probability Distribution and when is it used? When you have a finite population the hypergeometric probability distribution is better than the binomial distribution.

Meier & Brudney: Chapter 10 Special Probability Distributions (Page 2)  Let’s look at the Andersonville City Fire department example (pp ). There are 34 women out of a pool of 100 eligible applicants. The p =.34, and 30 trials (n = 30). Thus your mean is:  = 30 x.34 = The key difference is in computing the standard deviation.  The formula is shown on page 158 and the answer is on 159.

Meier & Brudney: Chapter 10 Special Probability Distributions (Page 3)  What is The Poisson distribution? It is named for the Frenchman, Simeon Dennis Poisson, who developed it in the first half of the 19th century. It is used to describe a pattern of behavior when some event occurs at varying, random intervals over a continuum of time, length, or space. Your text uses muggings & potholes as examples. It has been widely used to describe the probability functions of phenomena such as product demand; demand for services; # of calls through a switchboard; passenger cars at toll stations, etc.

Meier & Brudney: Chapter 10 Special Probability Distributions (Page 4)  How is the Poisson distribution different from the Bernoulli process? The number of trials is not known in a Poisson process. Instead, the Poisson distribution is concerned with a discrete, random variable which can take on the values x = 0,1,2,3... where the three dots mean "ad infinitum". The formula is:  = xe-/x! (see page 160)

Meier & Brudney: Chapter 10 Special Probability Distributions (Page 5)  The formula is:  = xe-/x! where:  (lambda) = the probability of an event  x = # of occurrences  e = (a constant that is the base of the Naperian or natural logarithm system)  Fortunately, this is not something we need to calculate; we'll use Table 2 (pp ), as we are sane.

Meier & Brudney: Chapter 10 Special Probability Distributions (Page 6)  Let’s look at the management example on pages  Why does the Poisson table only go to 20 for the λ values? We can use the normal distribution table for those values above 20.

Meier & Brudney: Chapter 10 Special Probability Distributions (Page 7)  Problems 10.2, 10.4, 10.6, & 10.16