Business Statistics Unit 5

Probability Distributions

This unit discusses probability distributions and two common probability distributions that are often used in statistics:  the binomial and the Poisson distribution.  These distributions are known as discrete probability distributions indicating that they have a set of limited values.

Introduction

What is a probability distribution?  It is simply a set of possible outcomes with each possible outcome having a probability of occurring.  In unit 4 we discussed basic probabilities and calculated the probability of an event occurring.  When we have a probability distribution we can take the probability of each event occurring and calculate the mean.  We simply use the concepts we learned in unit 3 to calculate the mean for a set of possible outcomes with each outcome having a specific probability of occurring.

For example, when you take a test, there might be five basic outcomes:  A, B, C, D, F.  However from the past we might have the following information by grade for all of the students taking the test last year.

As we look at the above probability distribution we can see that 110 out of 246 total students received an A.  Using the formula from unit 4 for the probability for a single event, we get 110/246 = .4472.  This is how you can simply interpret a probability distribution by each possible outcome.

The Mean, Variance, and Standard Deviation - Probability Distributions

The mean of a probability distribution is known as its expected value.  In this case the mean becomes a weighted average since each variable that is used to calculate the mean is weighted by its probability of occurrence.  To find the mean or expected value of a probability distribution we use the formula indicated below in formula box 5.1.

Formula Box 5.1.  The Mean or Expected Value of a Probability Distribution.

In this formula, P(x) is the probability of each value.  To find the mean we take each variable x and multiply it by its probability and then add all of them together.

To understand how to compute a mean for a probability distribution lets take a look at an example in Practice Exercise 5.1.

Practice Exercise 5.1.  Finding the Mean of a Probability Distribution.

A major appliance store in town is open six days a week.  The manager of the store has noticed that sales of televisions vary by day but generally follow the probability distribution given below for Saturdays.

What is the mean of the probability distribution?  To find the mean we apply formula 5.1 by taking each possible number of TVs sold, multiplying it by its probability, and adding the results together. (0 X .1) + (1 X .1) + (2 X .2) + (3 X .3) + (4 X .2) + (5 X .1) = 2.7 The manager can expect to sell on average 2.7 (roughly 3) televisions every Saturday.

The mean is only one possible calculation that we can use to describe a probability distribution.  The next calculation we need to learn is the variance.  The variance is used to describe how scattered or spread the data is within a probability distribution.  The formula for the variance is given below in Formula Box 5.2.

Formula Box 5.2.  The Variance of a Probability Distribution.

To use this formula you subtract from each variable x, the mean of the probability distribution, square the result, and then multiply by the probability of each variable x.  Lets use the formula to determine the variance for Practice Exercise 5.1.

Practice Exercise 5.2.  The Variance of a Probability Distribution.

In Practice Exercise 5.1 we examined sales of televisions at an appliance store on Saturdays.  The probability distribution is given below.

We found that the mean of the probability distribution was 2.7.  Now we need to calculate the variance.  Use formula 5.2 and follow the steps given below to find the variance.

The first step is to take each variable x and subtract the mean from each variable.

The next step is to square each value.

The next step is to multiply each squared value by its probability.

Our fourth and final step is to add all of the values in the fifth column together.  The values .729 + .289 + .098 + .027 + .338 + .529 = 1.9857.  The variance is equal to 2.01.

The next calculation to discuss is the standard deviation of a probability distribution.  As we discussed in a previous unit, the standard deviation is the square root of the variance.  The formula for the standard deviation of a probability distribution is given below in Formula Box 5.3.

Formula Box 5.3.  The Standard Deviation of a Probability Distribution.

To use this formula you subtract from each variable x, the mean of the probability distribution, square the result, and then multiply by the probability of each variable x.  The last step is to take the square root.  You should notice by now that the standard deviation formula is identical to the variance formula except an additional step is added at the end where the square root is taken.

Looking at Practice Exercises 5.1 and 5.2 we found the mean and variance for a probability distribution.  To find the standard deviation we simply take the square root of the variance.  The variance in Practice Exercise 5.2 is 2.01.  To find the standard deviation we simply take the square root of 2.01.  The standard deviation is 1.4177.

Can Excel do anything to help us here?  Yes!  You can enter all of the steps for the variance into Excel 2007 or 2003.  You can also find the standard deviation using Excel.  Let's take a look at the previous problem and use Excel to find our answers.

What we can do is add another column that contains a formula which includes all of the steps needed to calculate the variance.  To do this you simply enter the formula below in each cell.  If you copy the table into Excel starting at position A1, then in cells C2 through C7 you would need to enter the formula.  From our previous calculations, we know that the mean was equal to 2.7.

The last formula you need to enter is right below all of the C column formulas you just entered.  This formula adds up each calculated value to determine the variance.  Please note that you will only need three columns not four which are shown.  In order to show you both the formula and the calculated values for each cell it was necessary to use four columns.

Now that you have the variance, if you need to find the standard deviation you simply use the formula below in a blank Excel cell.

=SQRT(2.01)

You could also insert the cell address instead of 2.01 to find the standard deviation.  You should have answer of 1.417744688.

Binomial Distributions

A binomial probability distribution is a discrete probability distribution where there are only two possible outcomes.  The two possible outcomes are also mutually exclusive indicating that both possible answers cannot occur at the same time.  Examples of binomial distributions include true/false tests, flipping a coin, or a quality control manager that either accepts or rejects a shipment of raw materials used in production.

Binomial distributions involve counting possible outcomes.  For example if there are 10 true/false questions on a test, we count how many times the answer is true.  Or a quality control manager examines 20 shipments of raw materials and measures how many are acceptable.

A third characteristic of  binomial distributions is that they have probabilities that do not change from one trial to another.  Going back to our true/false test example, for the first test question you have a probability of .5 of getting the right answer.  You have the same probability of .5 getting the answer correct for the second question, and so on.

The fourth and final characteristic of binomial distributions is that each trial is independent of any other trial.  The first answer on a true/false test does not affect the outcome of the second answer and so on.  Looking at our quality control example, if you reject the first shipment received from a supplier, the rejection of that shipment does not influence whether or not you will reject the next shipment you examine from another supplier.

The formula for calculating a binomial distribution is given below in Formula Box 5.4.

Formula Box 5.4. 

P(r) = nCrprqn-r

To understand the formula for a binomial we must break it down.  The first part nCr is called a combination.  A combination is a selection of r items from a set of n distinct objects without regard to the order in which the r items are chosen.  A combination has its own formula which is given below.

 nCr   =     n! / r!(n - r)!

This formula uses factorials.  A factorial is simply all of the values at and below a variable.  For example, 5! is equal to 5 X 4 X 3 X 2 X 1 = 120.  4! is equal to 4 X 3 X 2 X 1 = 24.

The formula in box 5.4 also contains the symbol p which stands for the probability of success.  Some statistic books use the symbol π (pie) to refer to the probability of success.  The symbol q refers to the probability of failure and in some books it is shown as (1 - π ) or (1 - p).  The symbol n is the total number of items or objects and r is the number of items chosen from n (sometimes called x).

You can find the mean of any binomial distribution by taking the total number of items or objects (n) and multiplying it by the probability of success (p or π).  Therefore the mean or expected value of a binomial distribution is equal to:

µ = np

The variance (σ2) of a binomial distribution is found by multiplying the total number of objects (n) by the probability of success (p or π), and by the probability of failure (q or 1 - p or 1 - π).  Therefore the formula to find the variance is:

 σ2 = npq

For the standard deviation of a binomial distribution, we simply take the square root of the variance.

 σ = √npq

Instead of using formulas to solve binomial problems, a table of values has been developed to make the calculation of binomial probabilities easier.  In your textbook you will find a table of values for binomial probability distributions in the back of the book.  For some tables, the row at the top of the table provides different possible probabilities for p or π.  The two columns on the left side are labeled n and r.  Within the table are values based upon the value of n, r, and p.  Your tables may look different then the one described here.  Sometimes tables are arranged by the value of n first, with the probabilities given across the top and the value of r (sometimes called x) in a column on the left side.

Table 5.5.  Sample of Binomial Probability Distribution Table Values

Table 5.6.  Sample of Binomial Probability Distribution Table Values - Different Format

Lets use the table values to solve a typical binomial probability distribution problem.

Practice Exercise 5.3.  Using the Binomial Tables.

To solve this problem we must first determine our probability of success which is 10% or .10.  Then we must find our value of n, the number of trials, which is equal to 3 for this exercise.  Next we find the value of r (or x) which is 1.  Now we are ready to use the table.  You can use either table listed above to find the answer.  Using the first binomial table, we need to find p = .10, n = 3, and r = 1.

The correct answer is .2430.  There is a 24.3% chance that out of three locks selected, one will be defective.

The binomial tables provide a quick way to find a binomial probability.  You can also use Excel to find a binomial probability which is the preferred method for this course.

Practice Exercise 5.4.  Using Excel 2007 or 2003 to Find a Binomial Probability.

A quiz with five questions is given to a class of statistics students.  The quiz contains all True/False questions.  What is the probability of correctly guessing four of the five questions? 

For this exercise we will let Excel do the work for us following the steps indicated below.

Excel 2007

1. Make sure you have opened Excel and have a blank spreadsheet.  Move your cursor to column C1.

2. From the HOME folder, click on the Insert Function (fx) button.
   
   

3. Under the Function menu, go to the Select a category box, and scroll until you find Statistical.

4. Select Statistical and move your curser to the box called Select a function.

5. From the list of possible functions scroll until you find Binomdist.  Highlight Binomdist and then click OK.

6. The Function Arguments box should appear next.  Here is where you enter information on n, r, and p.

7. In the Number box, enter the number of successes which is 4.

8. In the Trials box, enter the number of trials which is 5.

9. In the Probability box, enter the probability of a success.  For each question, since there are two possible answers, the probability of one answer being correct is .50.  Therefore p = .50 so enter .50 into the Probability box.

10. In the Cumulative box, enter the number 0.
    
    

11. Now click OK.

12. Your answer which appears in cell C1 should be 0.15625.  That's it!

Excel 2003

1. Make sure you have opened Excel and have a blank spreadsheet.  Move your cursor to column C1.

2. Now click on Insert and then Function.

3. Under the Function menu, go to the Select a category box, and scroll until you find Statistical.

4. Select Statistical and move your curser to the box called Select a function.

5. From the list of possible functions scroll until you find Binomdist.  Highlight Binomdist and then click OK.

6. The Function Arguments box should appear next.  Here is where you enter information on n, r, and p.

7. In the Number box, enter the number of successes which is 4.

8. In the Trials box, enter the number of trials which is 5.

9. In the Probability box, enter the probability of a success.  For each question, since there are two possible answers, the probability of one answer being correct is .50.  Therefore p = .50 so enter .50 into the Probability box.

10. In the Cumulative box, enter the number 0.

11. Now click OK.

12. Your answer which appears in cell C1 should be 0.15625.  That's it!

Once again Excel makes our work easy.  Lets complicate things a bit by doing another practice exercise that involves a more complex problem.

Practice Exercise 5.5.  Using Excel 2007 or 2003 to Find a Binomial Probability - Complex Problems.

We are going to look at our quiz example again, but this time each question has four possible answers, and there are 10 questions on the quiz.  What is the probability of getting at least 7 questions correct?

There is an easier way to calculate Binomial probabilities when a question asks for a number of probabilities that are greater or less than a number value.  In this exercise we were asked to find the probabilities of getting at least 7 answers correct.  We can use Excel and the Binodist function and instead of entering each value of n separately, we simply use the Cumulative option differently.  If the cumulative box contains a value of 1 instead of 0, Excel will calculate the total probability of success from 0 to whatever value you have entered for n.  For example, if the number 6 is entered for n, and the cumulative box contains a 1 in Excel, Excel will calculate the total probability of success for 0, 1, 2, 3, 4, 5, and 6 successes combined.

We can use this feature to quickly find the answer to our problem in Practice Exercise 6.5.  Using the Binodist function, in the Function Arguments box, you enter 6 for the number of successes.  Why 6 and not 7?  Because Excel will calculate the total probabilities from 0 to a value of n, not above.  Therefore if we can find the probability of success for 0 through 6 successes, we simply subtract the total probability of 0 through 6 from 1.0 and we will know our probability for 7, 8, 9, and 10 successes.

Excel 2007

1. Move your cursor to C1 on a blank spreadsheet.

2. From the Home folder, lick on the Insert Function (fx) button.

3. Under the Function menu, go to the Select a category box, and scroll until you find Statistical.

4. Select Statistical and move your curser to the box called Select a function.

5. From the list of possible functions scroll until you find Binomdist.  Highlight Binomdist and then click OK.

6. The Function Arguments box should appear next.  Here is where you enter information on n, r, and p.

7. In the Number box, enter the number of successes which is 6.

8. In the Trials box, enter the number of trials which is 10.

9. In the Probability box, enter the probability of a success.  For each question, since there are four possible answers, the probability of one answer being correct is .25.  Therefore p = .25 so enter .25 into the Probability box.

10. In the Cumulative box, enter the number 1.

11. Now click OK.

12. Your first answer appears in cell C1 and it should be .996494.  This is the probability of getting 0 through 6.

13. To get the probability of 7 or more, subtract .996494 from 1.  We get .003506 our final answer.

Excel 2003

1. Move your cursor to C1 on a blank spreadsheet.

2. Now click on Insert and then Function.

3. Under the Function menu, go to the Select a category box, and scroll until you find Statistical.

4. Select Statistical and move your curser to the box called Select a function.

5. From the list of possible functions scroll until you find Binomdist.  Highlight Binomdist and then click OK.

6. The Function Arguments box should appear next.  Here is where you enter information on n, r, and p.

7. In the Number box, enter the number of successes which is 6.

8. In the Trials box, enter the number of trials which is 10.

9. In the Probability box, enter the probability of a success.  For each question, since there are four possible answers, the probability of one answer being correct is .25.  Therefore p = .25 so enter .25 into the Probability box.

10. In the Cumulative box, enter the number 1.

11. Now click OK.

12. Your first answer appears in cell C1 and it should be 0..996494.  This is the probability of getting 0 through 6.

13. To get the probability of 7 or more, subtract .996494 from 1.  We get .003506 our final answer.

Remember to subtract from 1 the probability you get from Excel if you use the cumulative value of 1 and are trying to find values greater than a stated number.  If you are trying to find the probability for values less than a stated number, you do not subtract from 1.

Poisson Probability Distributions

A Poisson probability distribution is used to determine the probability of a specified event occurring during a specific period of time.  The probability of an event occurring must be the same for each unit of time.  In addition, the number of occurrences in one time interval does not affect other intervals.

The Poisson distribution is often used to determine the probability of a defect being in a manufactured item like a computer or a car.  It can be used to determine to predict the number of insurance claims that will be received by an insurance company during a specified period of time.

The formula for the Poisson distribution can be found below in formula box 5.5.

Formula Box 6.5.  The Poisson Distribution.

P(x) = the probability of exactly x number of occurrences

µ = the mean number of occurrences over time

e = a constant value (2.71828)

x = the number occurrences (successes)

Other characteristics of the Poisson distribution include that the variance is equal to the mean.  An extension of this relationship is that the standard deviation is equal to the square root of the mean:  σ = √µ.  To find the mean of a Poisson distribution you simply multiply the number of trials by the probability of success.  This is denoted as:

µ=np or µ=nπ

where n is equal to the total number of trials, and p or π is equal to the probability of success.

The formula can be used to solve for Poisson probabilities but it is far easier to either use the Poisson tables or use Excel.  In the next practice exercise we will take a look at a common Poisson application and use the tables to solve the problem.

Table 5.7.  A Poisson Probability Table.

Practice Exercise 5.6.  Using the Poisson Tables to Solve a Problem.

General Motors has an auto assembly plant in Lansing Michigan which produces several types of cars including Cadillacs.  Past quality control evaluations have concluded that out of every 100 cars produced during a shift, 1 will have a significant defect when it's assembly is complete.  What is the probability of having 0 defects, 1 defect, or 2 defects during a given shift?

To solve this problem, we must first find the mean.  The mean µ is equal to the number of trials multiplied by the probability of success.  In our exercise the number of trials is equal to 100.  The probability of success or of the event occurring is 1 time out of 100 or .01.  Therefore our mean is equal to 100(.01) or 1.

Now we need to find the values for x in order to use the tables.  Our problem asked for the probability of having 0, 1, or 2 defects.  Therefore each of these represents the value of x.  So lets first use the Poisson table and find the probability for an x value of 0 and a mean of 1.  Go to the µ column 1.0 and the x row which is 0.  The probability is equal to .3679 that there will be 0 defects in cars being produced during a given shift.

Now we need to find the probability that there will be one car with a significant defect during a shift.  Here our x values equal 1 and the mean continues to be 1.0.  Using our table with µ = 1 and x = 1, we find that the probability of having one car with a significant defect is equal to .3679.

Our last step is to find the probability that during a shift there will be two cars with significant defects.  Once again our mean is equal to 1.0, but now our value of x is equal to 2.  Using the table, the probability of having two cars with significant defects is equal to .1839.

We have solved our problem!  The probability of having no defective cars on a shift is .3679, the probability of having one defective car is also .3679, and the probability of having two defective cars is .1839.  The Poisson distribution enables us to determine if our current production is meeting previous production standards.  For example, if during one shift six cars had significant defects, we know that the probability of this occurring is extremely low (.0005) which indicates that there is likely to be a new problem that is occurring during the assembly process.

Although the tables simplify the process used to calculate Poisson probabilities, you can also use Excel to calculate Poisson probabilities.  In the practice exercise below we will use Excel to find the Poisson probabilities.

Practice Exercise 5.7.  Using Excel 2007 or 2003 to Find Poisson Probabilities.

The city of Butte Montana is considering installing a traffic light at an intersection that currently has a two way stop.  The number of cars using this intersection were counted over a four week time period.  It was found that during any given week, an average of 3000 cars passed through this intersection.  The number of accidents were tracked also during the four week time period and it was found that an average of 15 accidents occurred.  Find the following probabilities:

That 15 accidents will occur.

That 5 accidents will occur.

Excel 2007

1. Make sure you have opened Excel and have a blank spreadsheet.  Move your cursor to column C1.

2. From the Home folder, click on the Insert Function (fx) button.

3. Under the Function menu, go to the Select a category box, and scroll until you find Statistical.

4. Select Statistical and move your curser to the box called Select a function.

5. From the list of possible functions scroll until you find Poisson.  Highlight Poisson and then click OK.

6. The Function Arguments box should appear next.  Here is where you enter information on x and the mean µ.

7. In the x box, enter the number of successes or occurrences which is 15 for the first question.

8. In the Mean box, enter the value of the mean.  Remember that you may need to calculate this using the formula µ=np.  The value of the mean for this problem is 3000 (.005) = 15.  The value p was found by taking 15/3000 = .005.

9. In the Cumulative box, enter the number 0.
   
   

10. Now click OK.

11. Your first answer appears in cell C1 and it should be .102436.

12. Now we need to answer the second question.  Move your cursor within Excel to position C2.

13. From the Home folder, click on the Insert Function (fx) button.

14. Under the Function menu, go to the Select a category box, and scroll until you find Statistical.

15. Select Statistical and move your curser to the box called Select a function.

16. From the list of possible functions scroll until you find Poisson.  Highlight Poisson and then click OK.

17. The Function Arguments box should appear next.  Here is where you enter information on x and µ.

18. In the x box, enter the number of successes or occurrences which is 5 for the second question.

19. In the Mean box, enter the value of the mean.  Remember that you may need to calculate this using the formula µ=np.  The value of the mean for this problem is 3000 (.005) = 15.  The value p was found by taking 15/3000 = .005.

20. In the Cumulative box, enter the number 0.

21. Now click OK.

22. Your second answer appears in cell C2 and it should be .001936.

Excel 2003

1. Make sure you have opened Excel and have a blank spreadsheet.  Move your cursor to column C1.

2. Now click on Insert and then Function.

3. Under the Function menu, go to the Select a category box, and scroll until you find Statistical.

4. Select Statistical and move your curser to the box called Select a function.

5. From the list of possible functions scroll until you find Poisson.  Highlight Poisson and then click OK.

6. The Function Arguments box should appear next.  Here is where you enter information on x and the mean µ.

7. In the x box, enter the number of successes or occurrences which is 15 for the first question.

8. In the Mean box, enter the value of the mean.  Remember that you may need to calculate this using the formula µ=np.  The value of the mean for this problem is 3000 (.005) = 15.  The value p was found by taking 15/3000 = .005.

9. In the Cumulative box, enter the number 0.

10. Now click OK.

11. Your first answer appears in cell C1 and it should be .102436.

12. Now we need to answer the second question.  Move your cursor within Excel to position C2.

13. Now click on Insert and then Function.

14. Under the Function menu, go to the Select a category box, and scroll until you find Statistical.

15. Select Statistical and move your curser to the box called Select a function.

16. From the list of possible functions scroll until you find Poisson.  Highlight Poisson and then click OK.

17. The Function Arguments box should appear next.  Here is where you enter information on x and µ.

18. In the x box, enter the number of successes or occurrences which is 5 for the second question.

19. In the Mean box, enter the value of the mean.  Remember that you may need to calculate this using the formula µ=np.  The value of the mean for this problem is 3000 (.005) = 15.  The value p was found by taking 15/3000 = .005.

20. In the Cumulative box, enter the number 0.

21. Now click OK.

22. Your second answer appears in cell C2 and it should be .001936.

Looking again at Practice Exercise 5.7, what if we wanted to find the probability of having 5 defects or less?  Excel makes this really easy to do.  In the Function Arguments box, you enter 5 for x, you still enter 15 in the Mean box, but when you get to the Cumulative box, you enter 1 instead of 0.  This will give you a total of all of the probabilities from 0 to 5.  The answer is .002792.  The cumulative function works the same way for both Binomial and Poisson distributions. 

Assignment

The homework problems for unit 5 can be found by going to the Course Documents link in Blackboard, and clicking on the link for Unit 5.  Look for the Unit 5 - Assignment link.  Once you have completed your homework assignment, you will need to post your answers on Blackboard®.  Once you have signed into Blackboard, simply go to the Unit 5 - Assignment 5 - Post Answers Here link and post your answers.  Immediate feedback is provided once you have completed the posting of all of your answers and clicked on submit.  Make sure you print the entire submitted homework assignment to assist you with quizzes and tests.

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