LAB SESSION 6

RANDOM NUMBERS AND PROBABILITY

INTRODUCTION:  This lab session is designed to introduce you to random numbers and their use in simulating experiments.   The outcomes of events in normal life cannot be predicted, but it is possible to have an idea of what outcomes are possible.  The theory of probability was developed to help analyze experiments whose outcomes are uncertain.  We can use Excel to simulate certain experiments such as flipping a coin or rolling a die.

RANDOM NUMBERS

You were introduced to the RANDOM command in Lab 5.  There we used the

RAND worksheet function, to return an evenly distributed random number greater than or equal to 0 and less than 1 every time the worksheet is calculated.  Now we’ll look at the Random Number Generation analysis tool.  This tool is part of the Analysis ToolPak.  This tool fills a range with independent random numbers drawn from one of several distributions.  You can characterize subjects in a population with a probability distribution.  For example, you might use a normal distribution to characterize the population of individuals’ heights, or you might use a Bernoulli distribution of two possible outcomes to characterize the population of coin toss results.

Suppose we want to simulate the outcomes for tossing a coin 100 times.

Choose:  Tools > Data Analysis > Random Number Generation > OK

Enter:      Number of Random Numbers:  100

Distribution:  Bernoulli

p-value:  0.5

Select:     Output Range:  select cell > OK

Give the relative frequency for a head (1) and a tail (2) based on the Excel output.  Certainly, let the computer do the work:

Enter a 0 and a 1 into the first two cells of column B.  (This is done to indicate the classes of values to be tallied.)  Select cells C1:C2 (to store the tallies) and then continue by typing

=Frequency(A1:A100, B1:B2)

Since this is an array formula you must press Ctrl + Shift + Enter

 Bins Frequency 0 46 1 54

Questions:

1.  What commands would be used for simulating the rolling of a die 50 times?

2.  Create a new Excel workbook and place 50 simulated rolls into columns A

and B.  Give the relative frequency for the outcomes 1, 2, 3, 4, 5, and 6 based

on the Excel output.

THE LAW OF LARGE NUMBERS

To see how the law of large numbers works, we need to create a third column with the sums of two dice rolls simulated by columns A and B.  First, since Excel generates real decimal values, place the INT (the integer function) of column A values in column C, and the INT of column B values into column D.

in cell E1 enter  = C1 + D1

click and drag the fill handle to cell E50

To determine the relative frequency of each outcome:

Enter: the possible outcomes 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 into column F

Select cells G1 through G11, and type =FREQUENCY(E1:E50,F1:F11) and

press Ctrl + Shift + Enter.  To get the relative frequencies, in cell H1 enter

=G1/50  and click and drag the fill handle to cell H11.

Note:  You may also use Data > Pivot Table Report . . .  to generate a table

 first roll second roll sum Bins frequency rel. freq. 2 3 5 2 0 0 3 4 7 3 2 0.04 4 2 6 4 7 0.14 5 4 9 5 5 0.1 1 5 6 6 10 0.2 2 4 6 7 10 0.2 1 5 6 8 5 0.1 4 5 9 9 7 0.14 1 2 3 10 3 0.06 2 5 7 11 0 0 1 4 5 12 0 0 3 1 4 5 3 8 1 3 4 ... ... ...

Interpreting the results:

1) What is the observed probability of obtaining a sum of 2 on the dice?

2) What is the observed probability of obtaining a sum of 7 on the dice?

3) What is the observed probability of obtaining a sum of 11 on the dice?

Using similar commands, create two additional columns containing 500 simulated rolls of a single die and a third column containing the sums of these 500 simulated rolls of 2 dice.

THE BINOMIAL PROBABILITY DISTRIBUTION

Consider the following situation: Suppose you bought four light bulbs.  The manufacturers claim that 85% of their bulbs will last at least 700 hours.  If the manufacturer is right, what are the chances that all four of your bulbs will last at least 700 hours?  That three will last 700 hours, but one will fail before that?

Consider another situation.  You've somehow gotten enrolled in a class in advanced Greek  Mythology.  You don’t know anything about mythology  but you’re to take a pop quiz.  You'll have to guess on every question.  It's a multiple-choice test; each of the 20 questions has 3 possible answers.  To pass you must get at least 12 correct.  What are the chances that you'll pass?

How would you answer the above questions?  Excel can help us with this by using the BINOMDIST (The Binomial Probability Distribution Function) to generate binomial probabilities.  (Remember what a binomial distribution requires.)

Calculating Binomial Probabilities with BINOMDIST

To obtain the probability of each possible outcome for a binomial distribution with n = 10 and p = 0.1, you will use the following commands.  You must first create a column with the values for which you wish to find the corresponding probabilities.  Input the values 0 to 10 into column A.  Activate B1, then continue with:

Choose:  Insert function, fx > Statistical > BINOMDIST > OK

Enter:     Number_s:              A1:A11, or select cells

Trials:                     10

Probability_s:  0.1

Cumulative:    false > OK

Drag:      fill handle down to give other probabilities

 A B 1 0 0.348678 2 1 0.38742 3 2 0.19371 4 3 0.057396 5 4 0.01116 6 5 0.001488 7 6 0.000138 8 7 8.75E-06 9 8 3.65E-07 10 9 9E-09 11 10 1E-10

This results in the following values:

Looking back to our original questions, to find the probability that three of your four light bulbs will be successes (last more than 700 hours) and one will fail we use:

“x-values” into column C (0, 1, 2, 3, 4) then activate cell D1 and

Choose:  Insert function, fx > Statistical > BINOMDIST > OK

Enter:     Number_s:              C1:C5, or select cells

Trials:                4

Probability_s:  0.85

Cumulative:    false > OK

Drag:      fill handle down to give other probabilities

 C D 1 0 0.000506 2 1 0.011475 3 2 0.097538 4 3 0.368475 5 4 0.522006

So we see that the probability of exactly 3 of the 4 bulbs being successes (lasting more than 700 hours) is .368475.

Cumulative Probabilities

The same statistical function BINODIST can be used to generate cumulative probabilities.  A cumulative probability is the probability that your result will be less than or equal to a particular value.  As an example, suppose we calculate the probability you will fail the test in advanced Greek Mythology.  Here n = 20 and p = .3333.  You will fail the test if you get less than or equal to 11 questions correct.  (You will pass if you get 12 or more right.)  The following commands can be used to calculate this probability:

Choose:  Insert function, fx > Statistical > BINOMDIST > OK

Enter:     Number_s:              A1:A21, or select cells

Trials:                     20

Probability_s:  1/3

Cumulative:    true > OK

Drag:      fill handle down to give other probabilities

The cell directly next to the x-value 11 is the probability you will fail (answer 11 or fewer correct).  So, what is the probability that you will pass?  ( 1 - .9870 = .013)

Mean and Standard Deviation of the Binomial Distribution

Excel can be used like a calculator to determine the mean and standard deviation for the binomial distribution.  Let’s continue using the Greek Mythology test example:

Activate cell C1, then

Enter:  =A1*B1

Click and drag:  fill handle down to complete the column calculation

Activate cell D1, then

Enter:  =A1*C1

Click and drag:  fill handle down to complete the column calculation

Activate cell C22 and click AutoSum and press enter (this is the mean)

Activate cell D22 and click AutoSum and press enter

(this is the sum of x2P(x))

Finish the calculation for standard deviation as follows:

Activate any cell and enter  =SQRT(D22-C22^2)

 mean std dev'n 6.666667 2.108185

6.67 is the mean number of questions expected to be answered correctly and 2.108 is the standard deviation expected among the number of questions answered correctly per test consisting of 20 questions.

ASSIGNMENT:   Do Exercises 4.30, 5.26, 5.66 - 5.68 in your text.