CHAPTER 5faculty.tamucc.edu/jguardiola/Math 1442/JMP Labs... · CHAPTER 5 Discrete Probability Distributions Probability distributions maps probabilities with events at the sample

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CHAPTER 5

Discrete Probability Distributions

Probability distributions maps probabilities with events at the sample space. This mapping can be done

using a table, a mathematical formula or a graph. In this Chapter we are going to study a table that

defines a discrete probability distribution and the binomial probability distribution.

A probability distribution must fulfill the following two requirements.

∑ ( )

( )

Class Exercises: Practice some computations for the mean, variance and standard deviation:

1- Compute the mean, variance and standard deviation for the following frequency distribution.

Table 5.1

x P(x)

0 0.30

1 0.35

2 0.20

3 0.10

4 0.05

first, let’s open a new data table:

Figure 5.1

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then, right click at the heading of “Column 1”

Figure 5.2

click on the text box for “Column Name” and change the name to “x”, as follows:

Figure 5.3

click twice at the right of first column heading to open a new column

Figure 5.4

right click over “Column 2”, select “Column Info” and change the name to “P(x)”, then fill the following

information:

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Figure 5.4

click twice over the space next to the column entitled “P(x)” and open a new column

Figure 5.5

change the name to xP(x) and click over “Column Properties”

Figure 5.6

select “Formula” from the menu

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Figure 5.7

click over “Edit Formula”

Figure 5.8

click over the variable x, then choose the multiplication sign and click over the variable P(x), you should

see the following screen

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Figure 5.9

click “OK” on this screen, and click “OK” on the next screen, you will see the following table:

Figure 5.10

next, let’s add a third column that contain the square of x times P(x). First click twice (or right click and

select new column) on the space next to xP(x) and then right click over “Column 4” and select “Column

Info…” as follows

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Figure 5.11

change the name to to x^2P(x), select “Formula” from the drop down menu “Column Properties” and

click over “Formula”

Figure 5.12

then write the formula

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Figure 5.13

click “OK” in this window and the next window, you will see the results on the following window

Figure 5.14

Let’s compute the mean and expected value. In order to do that we need to compute the sum of the

columns labeled P(x) (for verification purposes), xP(x) and x^2P(x). Click over “Tables”, and select

“Tabulate” as follows:

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Figure 5.15

Next, drag the variables P(x), xP(x) and x^2P(x), one at a time, over the “Drop zone for columns” and

select “Add analysis column” for each variable, you will get the following table:

Figure 5.16

notice that the sum of P(x) is equal to 1 (requirement for a probability distribution). With these results

you can compute the mean, the variance and the standard deviation as follows:

µ = 1.25 and σ²=2.85 – 1.25² = 1.2875, then σ=√(1.2875) = 1.135

2- Compute binomial probabilities. Flip a coin 10 times, keeping track of the number of heads.

Compute a probability for each value of x.

First, open a new file and click over the first cell at the leftmost side (on the cell below the red

triangle) and select the option “Add Rows…”

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Figure 5.17

type 11 for the number of rows as follows, then click “OK”

Figure 5.18

then, you can see the following screen with dots on the corresponding empty spaces

Figure 5.19

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right click over Column 2, change the name on the dialog box to “Success” and click over

“Missing/Empty” and then over “Sequence Data”

Figure 5.20

input the numbers 0 on “From” and 10 on “To”, accept the default of 1 on “Step” and click over “OK”

Figure 5.21

you will get have the following sequence of numbers

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Figure 5.22

now, let’s assign probabilities to the corresponding number of successes. Double click at the right of the

column “Successes” and add a new column. Label it as P(x), and choose “Formula” from the menu

“Column Properties”, then click over “Edit Formula” as follows

Figure 5.23

from the list of functions choose “Discrete Probability” and then choose “Binomial Probability” as

follows:

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Figure 5.24

type 0.5 over “p”, type 10 over “n”, and select “Successes” for “x” as follows, then click “OK” on this

window and on the next window

Figure 5.25

You will get the following results

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Figure 5.26

These are the probabilities for the corresponding number of successes; you can verify that the sum of

the numbers at the first column is 1 as required but the theory. To do these computations choose

“Tables” and “Tabulate” as we did before (procedure not shown, please ask your instructor if you need

some help). Next, we would like to compute the cumulative probability at the next column, to do this,

double click at the space at the right side of “P(x)”, change the name for the new column to “Cumulative

Prob”, select “Formula” and “Edit Formula” as we have done before.

Figure 5.27

Then, select “Discrete Probability” from the list of functions, and “Binomial Distribution” from the drop

down menu as follows:

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Figure 5.28

fill the same information as before, 0.5 for p, 10 for n and choose the variable “Successes” for x, as

follows

Figure 5.29

click over “OK” on this window and the next window and you will see the results:

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Figure 5.30

you can see that the third column is the cumulative probability for the binomial distribution. The last

number at the list is 1, as expected (it is the sum of all the previous probabilities).

3- Simulate binomial probabilities for flipping a coin 10 times, keeping track of the number of

heads.

Open a new data table as described in Figure 5.1. Then right click over the shaded area to the left of

column and select add rows

Figure 5.31

type 1000 at the dialog box and click OK

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Figure 5.32

you will see many rows with a dot. Next, right click over the heading of “Column 1” and select “Column

Info” as in figure 5.2. Then change the name to “Simulation”, click over the black triangle at the button

“Column Properties” and select ‘Formula”, then click over “Edit Formula” as shown

Figure 5.33

click “OK”, then Select “Random” from “Functions (grouped)” and click over “Random Binomial”, type

the parameters of the simulation, 10 for “n” and 0.5 for “p” as shown

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Figure 5.34

then, you will see the results of the simulation (results may differ)

Figure 5.35

Here you have 1,000 simulations for 10 trials each, the numbers that you see there are the number of

successes for each experiment of 10 trials. Now, let’s summarize these results using the option “Table”

and select “Tabulate” from the drop down menu

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Figure 5.36

then, drag the variable “Simulation” over the “Drop Columns” area

Figure 5.37

you can see the results shown at the table (results may differ)

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Figure 5.38

you can compute the probabilities dividing these numbers by 1000 (the number of experiments), now

let’s compare the simulated results with the actual probabilities

Table 5.2

x P(x) Simulated prob

0 0.001 0.001

1 0.010 0.013

2 0.044 0.052

3 0.117 0.120

4 0.205 0.199

5 0.246 0.239

6 0.205 0.202

7 0.117 0.114

8 0.044 0.049

9 0.010 0.010

10 0.001 0.001

As you can see, the simulated results are very close to the theoretical probabilities computed using the

binomial formula.

Class Exercises:

1- Compute the mean and standard deviation for the following probability distribution.

Table 5.3

x P(x)

0 0.20

1 0.25

2 0.22

3 0.18

4 0.10

5 0.05

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2- A couple decides to have 5 Children. Compute the binomial probabilities for the number of girls

that they can have.

3- Generate a simulation for the previous problem. Compare the simulated results with the

computed probabilities from problem 2.

Team Assignment:

A county is 50% African-American and 50% Caucasian, and an African-American is on trial. A

jury has been selected that contains six Caucasians and two African-Americans (this is an 8

person jury). The defendant claims racial bias in the jury selection because its makeup

seems unlikely given the racial percentages in the county. However, the prosecutor claims

that the jury was selected without regard to race. Is the defendant’s claim credible? In

other words, is selecting 2 or fewer African-Americans unlikely to happen by chance? The

defendant needs to know these probabilities in order to see if he should pursue his claim. If

he pursues this claim and is found correct, he could get a new jury and a better chance at

acquittal. However, if he is found incorrect he will have spent more money for his lawyer to

pursue the claim and will have the same jury that has been chosen.

The defendant has hired your investigative team to investigate this claim. By your vast

statistical expertise you have decided to employ three methods to gain evidence to advise

the man on trial:

Method 1: Calculate a theoretical probability by using the binomial formula

Method 2: Perform a simulation in Minitab and compare results with the “true”

probabilities compute in Method 2.

You will compare these probabilities to each other, make a decision, and advise the man.

The question you need to answer:

Should the defendant pursue his claim of racial bias in jury selection or should he stick with

the jury that was selected because most likely it just happened by chance?

Make sure that you provide a solid statistical support for you decision.