Chapter 6

DiscreteProbability DistributionsChapter 6

Copyright 2015 McGraw-Hill Education.All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.1Learning ObjectivesLO6-1 Identify the characteristics of a probability distribution.LO6-2 Distinguish between discrete and continuous random variables.LO6-3 Compute the mean, variance, and standard deviation of a discrete probability distribution.LO6-4 Explain the assumptions of the binomial distribution and apply it to calculate probabilities.LO6-5 Explain the assumptions of the hypergeometric distribution and apply it to calculate probabilities.LO6-6 Explain the assumptions of the Poisson distribution and apply it to calculate probabilities.6-#2What is a Probability Distribution?PROBABILITY DISTRIBUTION A listing of all the outcomes of an experiment and the probability associated with each outcome.LO6-1 Identify the characteristics of a probability distribution.6-#3Characteristics of a Probability DistributionThe probability of a particular outcome is between 0 and 1 inclusive.The outcomes are mutually exclusive events.The list is exhaustive. So the sum of the probabilities of the various events is equal to 1.LO6-16-#4Probability Distribution - ExampleExperiment: Toss a coin three times. Observe the number of heads.

The possible experimental outcomes are: zero heads, one head, two heads, and three heads.

What is the probability distribution for the number of heads?LO6-1

6-#5Probability Distribution: Number of Heads in 3 Tosses of a Coin

LO6-16-#6Random Variables

RANDOM VARIABLE A quantity resulting from an experiment that, by chance, can assume different values.

LO6-2 Distinguish between discrete and continuous random variables.6-#7Types of Random VariablesDISCRETE RANDOM VARIABLE A random variable that can assume only certain clearly separated values. It is usually the result of counting something.

CONTINUOUS RANDOM VARIABLE A random variable that can assume an infinite number of values within a given range. It is usually the result of some type of measurement.

LO6-26-#8Discrete Random VariableEXAMPLES:The number of students in a classThe number of children in a familyThe number of cars entering a carwash in a hourThe number of home mortgages approved by Coastal Federal Bank last weekDISCRETE RANDOM VARIABLE A random variable that can assume only certain clearly separated values. It is usually the result of counting something.

LO6-26-#9Continuous Random VariableEXAMPLES:The length of each song on the latest Tim McGraw CDThe weight of each student in this classThe amount of money earned by each player in the National Football LeagueCONTINUOUS RANDOM VARIABLE A random variable that can assume an infinite number of values within a given range. It is usually the result of some type of measurement

LO6-26-#10The Mean of a Discrete Probability DistributionThe mean is a typical value used to represent the central location of a probability distribution.The mean of a probability distribution is also referred to as its expected value.LO6-3 Compute the mean, variance, and standard deviation of a discrete probability distribution.

6-#11The Mean of a Discrete Probability Distribution - ExampleJohn Ragsdale sells new cars for Pelican Ford. John usually sells the largest number of cars on Saturday. He has developed the following probability distribution for the number of cars he expects to sell on a particular Saturday.

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6-#12The Mean of a Discrete Probability Distribution - ExampleLO6-3

6-#13The Variance and StandardDeviation of a Discrete Probability DistributionMeasures the amount of spread in a distribution.

The computational steps are:Subtract the mean from each value, and square this difference.Multiply each squared difference by its probability.Sum the resulting products to arrive at the variance.

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6-#14The Variance and StandardDeviation of a Discrete Probability Distribution - Example

LO6-36-#15Binomial Probability DistributionA widely occurring discrete probability distributionCharacteristics of a binomial probability distribution:There are only two possible outcomes on a particular trial of an experiment.The outcomes are mutually exclusive.The random variable is the result of counts.Each trial is independent of any other trial.

LO6-4 Explain the assumptions of the binomial distribution and apply it to calculate probabilities.6-#16Characteristics of a Binomial Probability ExperimentThe outcome of each trial is classified into one of two mutually exclusive categoriesa success or a failure.

The random variable, x, is the number of successes in a fixed number of trials.

The probability of success and failure stay the same for each trial.

The trials are independent, meaning that the outcome of one trial does not affect the outcome of any other trial.LO6-46-#17Binomial Probability FormulaLO6-4

6-#18Binomial Probability - ExampleThere are five flights daily from Pittsburgh via US Airways into the Bradford Regional Airport. Suppose the probability that any flight arrives late is 0.20. What is the probability that none of the flights are late today?

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Recall: 0! = 1, and, any variable with a 0 exponent is equal to one.6-#19Binomial Distribution Probability LO6-4The probabilities for each value of the random variable, number of late flights (0 through 5), can be calculated to create the entire binomial probability distribution.

6-#20Mean and Variance of a Binomial DistributionLO6-4Knowing the number of trials, n, and the probability of a success, , for a binomial distribution, we can compute the mean and variance of the distribution.

6-#21For the example regarding the number of late flights, recall that =.20 and n = 5. What is the average number of late flights?What is the variance of the number of late flights?

Mean and Variance of a Binomial Distribution - Example

LO6-46-#22Mean and Variance of a Binomial Distribution Example

LO6-4Using the general formulas for discrete probability distributions:6-#23Binomial Probability Distributions TablesLO6-4Binomial probability distributions can be listed in tables. The calculations have already been done. In the table below, the binomial distributions for n=6 trials, and the different values of the probability of success are listed.

6-#24Binomial Probability Distribution Tables ExampleFive percent of the worm gears produced by an automatic, high-speed Carter-Bell milling machine are defective. What is the probability that out of six gears selected at random none will be defective? Exactly one? Exactly two? Exactly three? Exactly four? Exactly five? Exactly six out of six?

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6-#Binomial Probability Distribution- Excel ExampleLO6-4Using the Excel Function: Binom.dist(x,n,,false)

6-#26Binomial Probability Distribution MegaStat ExampleFive percent of the worm gears produced by an automatic, high-speed Carter-Bell milling machine are defective. What is the binomial probability distribution of the number defective when six gears are selected?

LO6-4Statistical software, such as Megastat, can also create the values and graph for any binomial probability distribution.6-#27Binomial Shapes or Skewness for Varying and n=10The shape of a binomial distribution changes as n and change.

LO6-46-#28Binomial Shapes or Skewness for Constant and Varying n

LO6-46-#29Binomial Probability Distributions Excel ExampleLO6-4A study by the Illinois Department of Transportation showed that 76.2 percent of front seat occupants used seat belts. If a sample of 12 cars traveling on a highway are selected, the binomial probability distribution of cars with front seat occupants using seat belts can be calculated as shown.

6-#30Binomial Probability Distributions Excel ExampleWhat is the probability the front seat occupants in exactly 7 of the 12 vehicles are wearing seat belts?

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6-#31Cumulative Binomial Probability Distributions Excel ExampleWhat is the probability the front seat occupants in at least 7 of the 12 vehicles are wearing seat belts?

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6-#32Hypergeometric Probability DistributionAn outcome on each trial of an experiment is classified into one of two mutually exclusive categoriesa success or a failure.The probability of success and failure changes from trial to trial. The trials are not independent, meaning that the outcome of one trial affects the outcome of any other trial.

Note: Use hypergeometric distribution if the experiment is binomial, but sampling is without replacement from a finite population where n/N is more than 0.05.LO6-5 Explain the assumptions of the hypergeometric distribution and apply it to calculate probabilities.6-#33Hypergeometric Probability Distribution - Formula

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6-#34Hypergeometric Probability Distribution - ExamplePlayTime Toys, Inc., employs 50 people in the Assembly Department. Forty of the employees belong to a union and ten do not. Five employees are selected at random to form a committee to meet with management regarding shift starting times.

What is the probability that four of the five selected for the committee belong to a union?

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6-#35Hypergeometric Probability Distribution - ExampleHeres whats given:N = 50 (number of employees)S = 40 (number of union employees)x = 4 (number of union employees selected)n = 5 (number of employees selected)What is the probability that four of the five selected for the committee belong to a union?

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6-#36Poisson Probability DistributionThe Poisson probability distribution describes the number of times some event occurs during a specified interval. The interval may be time, distance, area, or volume.

Assumptions of the Poisson Distribution:The probability is proportional to the length of the interval. The intervals are independent.

LO6-6 Explain the assumptions of the Poisson distribution and apply it to calculate probabilities.6-#37Poisson Probability DistributionThe Poisson probability distribution is characterized by the number of times an event happens during some interval or continuum. Examples:The number of misspelled words per page in a newspaperThe number of calls per hour received by Dyson Vacuum Cleaner CompanyThe number of vehicles sold per day at Hyatt Buick GMC in Durham, North CarolinaThe number of goals scored in a college soccer game

LO6-66-#38Poisson Probability DistributionThe Poisson distribution can be described mathematically by the formula:LO6-6

6-#39Poisson Probability DistributionThe mean number of successes, , can be determined in Poisson situations by n, where n is the number of trials and the probability of a success.

The variance of the Poisson distribution is also equal to n .

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6-#40Poisson Probability Distribution ExampleAssume baggage is rarely lost by Northwest Airlines. Suppose a random sample of 1,000 flights shows a total of 300 bags were lost. Thus, the arithmetic mean number of lost bags per flight is 0.3 (300/1,000). If the number of lost bags per flight follows a Poisson distribution with u = 0.3, find the probability of not losing any bags.

LO6-66-#41Poisson Probability Distribution Table Example Recall from the previous illustration that the number of lost bags follows a Poisson distribution with a mean of 0.3. A table can be used to find the probability that no bags will be lost on a particular flight. What is the probability no bag will be lost on a particular flight? LO6-6

6-#42More About the Poisson Probability DistributionThe Poisson probability distribution is always positively skewed and the random variable has no specific upper limit. The Poisson distribution for the lost bags illustration, where =0.3, is highly skewed. As becomes larger, the Poisson distribution becomes more symmetrical.LO6-6

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