WK3 DQ - Data Analysis & Business Intelligence

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Discrete Probability Distributions

Chapter 6

6-1

This chapter begins the study of probability distributions. We first discuss the mean, variance, and standard deviation of a probability distribution. Then we discuss three frequently occurring probability distributions: the binomial, hypergeometric, and Poisson.

Learning Objectives

LO6-1 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

6-2

What is a Probability Distribution?

Example: A drug manufacturer may claim a treatment will cause weight loss for 80% of the population. This claim could be tested by a consumer protection agency using a sample and statistical inference.

PROBABILITY DISTRIBUTION A listing of all the outcomes of an experiment and the probability associated with each outcome.

CHARACTERISTICS OF A PROBABILITY DISTRIBUTION

The probability of a particular outcome is between 0 and 1 inclusive.

The outcomes are mutually exclusive.

The list of outcomes is exhaustive. So the sum of the probabilities of the outcomes is equal to 1.

6-3

A probability distribution gives the entire range of values that can occur based on an experiment.

Probability Distribution Example

Suppose we are interested in the number of heads showing face up with 3 tosses of a coin

The possible outcomes are 0 heads, 1 head, 2 heads, and 3 heads

6-4

Using the counting formula from chapter 5, we note there are 8 possible outcomes when tossing a coin three times, since you will observe either a head or a tail, two possibilities, and this toss is repeated 3 times, so (2)(2)(2) = 8. Using this data we can construct a probability distribution.

Probability Distribution Table

Probability distribution table and chart for the events of zero, one, two, and three heads

6-5

Tossing the coin three times results in 8 different outcomes. The events of 0, 1, 2, or 3 heads are mutually exclusive and exhaustive so the sum of the probabilities is 1.000.

Random Variables

In any experiment of chance, the outcomes occur randomly, and so are called random variables

Examples

The number of employees absent from the day shift on Monday: the number might be 0, 1, 2, 3, …The number absent is the random variable

The grade level (Freshman, Sophomore, Junior, or Senior) of the members of the St. James High School Varsity girls’ basketball team. Grade level is the random variable (and notice that it is a qualitative variable).

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

6-6

In any experiment of chance, the outcomes occur randomly. So the outcome is called a random variable. Random variables can be measured with quantitative variables or qualitative variables.

Two Types of Random Variables

One type of random variable is the discrete random variable

Discrete variables are usually the result of counting

Example

Tossing a coin three times and counting the number of heads

DISCRETE RANDOM VARIABLE A random variable that can assume only certain clearly separated values.

6-7

Note: a discrete random variable can assume fractional or decimal values but the values must be separated by having distance between each value.

Discrete Random Variable

For example, the Bank of the Carolinas counts the number of credit cards carried by a group of customers

The number of cards carried is the discrete random variable

6-8

The results of this survey were obtained by counting the number of credit cards carried by a group of customers and are summarized in the table using relative frequencies.

Continuous Random Variables

Continuous variables are usually the result of measuring

Examples

The time between flights between Atlanta and LA are 4.67 hours, 5.13 hours, and so on

The annual snowfall in Minneapolis, MN measured in inches

6-9

CONTINUOUS RANDOM VARIABLE A random variable that may assume an infinite number of values within a given range.

Continuous random variables are measured with interval or ratio scale and the likelihood of a continuous random variable can be summarized with a probability distribution. This will be covered in the next chapter.

Mean and Variance of a Probability Distribution

The mean is a typical value used to represent the central location of the data

The mean is also referred to as the expected value

The amount of spread (or variation) in the data is described by the variance

The standard deviation of the probability distribution is the positive square root of the variance

6-10

The mean of a probability distribution is the long run average of the random variable. P(x) represents the probability of a particular value of x. To calculate the mean, multiply each value of x by its probability of occurrence and then add these products.

Probability Distribution Mean Example

John Ragsdale sells new cars for Pelican Ford. John usually sells the most cars on Saturday. He has developed a probability distribution for the number of cars he expects to sell on Saturday.

What type of distribution is this?

How many cars does John expect to sell on a typical Saturday?

What is the variance?

6-11

This is a discrete probability distribution. We find John can expect to sell 2.1 cars on a typical Saturday. In other words, over the long run, say 50 Saturdays in a year, he can expect to sell (50)(2.1) = 105 cars. The variance is calculated on the next slide.

Probability Distribution Variance Example

The computational steps for variance:

Subtract the mean from each value of x and square

Multiply each squared difference by its probability

Sum the resulting products to arrive at the variance

6-12

Take the (positive) square root of the variance to get the standard deviation. The mean is 2.1, the variance is 1.290, and the standard deviation is 1.136 cars.

Binomial Distribution

There are four requirements of a binomial probability distribution

There are only two possible outcomes and the outcomes are mutually exclusive, as either a success or a failure

The number of trials is fixed and known

The probability of a success is the same for each trial

Each trial is independent of any other trial

Example

A young family has two children, both boys. The probability of the third birth being a boy is still .50. The gender of the third child is independent of the gender of the other two.

6-13

The binomial distribution is a widely occurring discrete probability distribution.

Binomial Probability Experiment

BINOMIAL PROBABILITY EXPERIMENT

An outcome on each trial of an experiment is classified into one of two mutually exclusive categories — a success or a failure.

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

The probability of success is 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.

6-14

Use the number of trials, n, and the probability of a success, to compute binomial probability

Note: Do not confuse the symbol with the mathematical constant 3.1416

The Greek letter (pi) is used to denote a binomial population parameter. The formula to compute binomial probability is found on the next slide.

How is a Binomial Probability Computed?

Recently, www.creditcards.com reported that 28% of purchases at coffee shops were made with a debit card. For 10 randomly selected purchases at the Starbucks on the corner of 12th Street and Main:

P(x) = nCr(

P(0) = 10C0(

= (1)(1)(.0374) = .0374

P(x) = nCr(

P(1) = 10C1(

= (10)(25)(.0520) = .1456

What is the probability that no purchases were made with a debit card?

What is the probability that exactly one was made with a debit card?

6-15

C represents a combination; you may recall we calculated combinations in chapter 5. In this example, just insert the value for x that you wish to find, the value 5 for n and .20 for , and solve. With an n = 5, we can find the probabilities for 6 different x outcomes: 0, 1, 2, 3, 4, and 5 late flights. See the table on the next page.

Binomial Probability Distribution

Recently, www.creditcards.com reported that 28% of purchases at coffee shops were made with a debit card. For 10 randomly selected purchases at the Starbucks on the corner of 12th Street and Main, what is the probability that exactly one of the purchases was made with a debit card? What is the probability distribution for the random variable, number of purchases made with a debit card? What is the probability that six or more purchases out of 10 are made with a debit card? What is the probability that five or less purchases out of 10 are made with a debit card?

6-16

Use formula 6-3 to complete the table by finding the P(x) for the number of successes from 0 to 5. Then multiply each value of x by its probability and sum the products. The result will be the mean. Next, find the differences of each value of x and the mean and then square these differences. Multiply this squared value by the respective probability and sum the products, the result is the variance.

Shortcut Formulas

6-17

Using the preceding example of debit card purchases; n=10 and

Here is a shortcut method of finding the mean and variance of a binomial distribution. Verify these results with the table on the preceding slide.

Binomial Probability Tables

Tables are already constructed for use as well

In the Southwest, 5% of all cell phone calls are dropped. What is the probability that out of six randomly selected calls, none was dropped? Exactly one? Exactly two? Exactly three? Exactly four? Exactly five? Exactly six out of six? See the table below for the answers.

6-18

In this example, the conditions for a binomial distribution are met. You can build a binomial probability distribution using Formula 6-3 for any value of n and . However, the calculations take time. For convenience, tables like the one here are provided in Appendix B.1 for various values of n and .

Excel can be used to find binomials as well.

Cumulative Binomial Probability Distributions

A study by the Illinois Department of Transportation concluded that 76.2% of front seat occupants wore seat belts. That is, both occupants of the front seat were using their seat belts. Suppose we decide to compare that information with current usage. We select a sample of 12 vehicles.

What is the probability that the front seat occupants in exactly 7 of the 12 vehicles are wearing seat belts?

P(x) = nCr(

P(x=7) = 12C7(

= 792(.149171)(.000764) = .0902

What is the probability that at least 7 of the 12 front seat occupants are wearing seat belts?

P(x≥7) = P(x=7) + P(x=8) + P(x=9) + P(x=10) + P(x=11) + P(x=12)

=.0902 + .1805 + .2569 + .2467 + .1436 + .0383

=.9562

6-19

We conclude the likelihood that the occupants of exactly 7 out of 12 sampled vehicles will be wearing their seat belts is about 9%. And the probability of selecting 12 cars and finding that the occupants of 7 or more vehicles were wearing seat belts is .9562. The Excel commands for these calculations are found in Appendix C.

Hypergeometric Distribution

When sampling from relatively small populations without replacement, use the hypergeometric distribution

6-20

Here is a list of the characteristics of the Hypergeometric Distribution.

Hypergeometric Formula

PlayTime Toys Inc. employs 50 people in the Assembly Dept. Forty of the employees belong to a union and 10 do not. Five employees are selected at random to form a committee. What is the probability that four of the five belong to a union?

= = .431

6-21

Insert the following values in the formula and solve. N=50, S=40, x is 4, n is 5. Thus, the probability of selecting 5 assembly workers at random from the 50 workers and finding 4 of the 5 are union members is .431.

Hypergeometric Probabilities

6-22

Table 6-4 Hypergeometric Probabilities (n=5, N=50, and S=40) for the Number of Union members on the Committee. This table shows the hypergeometric probabilities of finding 0, 1, 2, 3, 4, and 5 union members on the committee. Excel can be used to create a hypergeometric distribution too; the steps to do so can be found in Appendix C.

Poisson Probability Distribution

This describes the number of times some event occurs during a specified interval

The interval can be time, distance, area, or volume

Two assumptions

The probability is proportional to the length of the interval

The intervals are independent

The Poisson has many applications, like describing:

The distribution of errors in data entry

The number of accidents on I-75 during a three-month period

6-23

The Poisson is a discrete probability distribution since it is formed by counting. It is often referred to as the law of improbable events because the probability of an event happening is small.

Poisson Distribution

6-24

The variance of the Poisson Distribution is equal to its mean.

Poisson Distribution Example

The probability that no bags are lost is found using formula 6-7.

= .9608

Then calculate the probability that one or more bags is lost.

P(x≥1) =1- 1 - = 1- .9608 = .0392

6-25

Budget Airlines is a seasonal airline that operates flights from Myrtle Beach, South Carolina, to various cities in the northeast. Recently Budget has been concerned about the number of lost bags. Ann Poston from the Analytics Department was asked to study the issue. She randomly selected a sample of 500 flights and found that a total of twenty bags were lost on the sampled flights.

The mean number of bags lost, is found by 20/500 = .04

Begin by confirming the situation fits the Poisson Distribution characteristics. Then calculate the mean using the sample information. Poisson probabilities can be found using Excel too; the commands can be found in Appendix C.

Poisson Probability Distribution Tables

NewYork-LA Trucking company finds the mean number of breakdowns on the New York to Los Angeles route is 0.30. From the table, we can locate the probability of no breakdowns on a particular run. Find the column 0.3, then read down that column to the row labeled 0; the value is .7408. The probability of 1 breakdown is .2222

6-26

Table 6-7 Poisson Table for Various Values of The Poisson distribution tables can be found in Appendix B.2

Chapter 6 Practice Problems

6-27

Question 5

6-28

The information below is the number of daily emergency service calls made by the volunteer ambulance service of Walterboro, South Carolina, for the last 50 days. To explain, there were 22 days when there were two emergency calls, and 9 days when there were three emergency calls.

Convert this information on the number of calls to a probability distribution.

Is this an example of a discrete or continuous probability distribution?

What is the probability that 3 or more calls are made in a day?

What is the mean number of emergency calls per day?

What is the standard deviation of the number of calls made daily?

LO6-2,3

Question 13

6-29

An American Society of Investors survey found 30% of individual investors have used a discount broker. In a random sample of nine individuals, what is the probability:

Exactly two of the sampled individuals have used a discount broker?

Exactly four of them have used a discount broker?

None of them has used a discount broker?

LO6-4

Question 21

6-30

In a recent study, 90% of the homes in the United States were found to have large-screen TVs. In a sample of nine homes, what is the probability that:

All nine have large-screen TVs?

Less than five have large-screen TVs?

More than five have large-screen TVs?

At least seven homes have large-screen TVs?

LO6-4

Question 25

6-31

A youth basketball team has 12 players on their roster. Seven of the team members are boys and five are girls. The coach writes each player’s name on a sheet of paper and places the names in a hat. The team captain shuffles the names and the coach selects five slips of paper from the hat to determine the starting lineup.

What is the probability the starting lineup consists of three boys and two girls?

What is the probability the starting lineup is all boys?

What is the probability there is at least one girl in the starting lineup?

LO6-5

Question 33

6-32

Ms. Bergen is a loan officer at Coast Bank and Trust. From her years of experience, she estimates that the probability is .025 that an applicant will not be able to repay his or her installment loan. Last month she made 40 loans.

What is the probability that three loans will be defaulted?

What is the probability that at least three loans will be defaulted?

LO6-6