# Mathematics

**11.College Education and Business World Success** R. H. Bruskin Associates Market Research found that 40% of Americans do not think that having a college education is important to succeed in the business world. If a random sample of five Americans is selected, find these probabilities.

*a*.Exactly 2 people will agree with that statement.

*b*.At most 3 people will agree with that statement.

*c*.At least 2 people will agree with that statement.

*d*.Fewer than 3 people will agree with that statement.

Source: *100% American* by Daniel Evans Weiss.

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**12.Destination Weddings** Twenty-six percent of couples who plan to marry this year are planning destination weddings. In a random sample of 12 couples who plan to marry, find the probability that

*a*.Exactly 6 couples will have a destination wedding

*b*.At least 6 couples will have a destination wedding

*c*.Fewer than 5 couples will have a destination wedding

Source: *Time* magazine.

**13.People Who Have Some College Education** Fifty-three percent of all persons in the U.S. population have at least some college education. Choose 10 persons at random. Find the probability that

*a*.Exactly one-half have some college education

*b*.At least 5 do not have any college education

*c*.Fewer than 5 have some college education

Source: *New York Times Almanac.*

**14.(ans)** Find the mean, variance, and standard deviation for each of the values of *n* and *p* when the conditions for the binomial distribution are met.

*a*.*n* = 100, *p* = 0.75

*b*.*n* = 300, *p* = 0.3

*c*.*n* = 20, *p* = 0.5

*d*.*n* = 10, *p* = 0.8

*e*.*n* = 1000, *p* = 0.1

*f*.*n* = 500, *p* = 0.25

*g*.*n* = 50, *p* =

*h*.*n* = 36, *p* =

**15.Social Security Recipients** A study found that 1% of Social Security recipients are too young to vote. If 800 Social Security recipients are randomly selected, find the mean, variance, and standard deviation of the number of recipients who are too young to vote.

Source: *Harper’s Index.*

**16.**Find the mean, variance, and standard deviation for the number of heads when 20 coins are tossed.

**17.Defective Calculators** If 3% of calculators are defective, find the mean, variance, and standard deviation of a lot of 300 calculators.

**18.Federal Government Employee E-mail Use** It has been reported that 83% of federal government employees use e-mail. If a sample of 200 federal government employees is selected, find the mean, variance, and standard deviation of the number who use e-mail.

Source: *USA TODAY.*

**19.Watching Fireworks** A survey found that 21% of Americans watch fireworks on television on July 4. Find the mean, variance, and standard deviation of the number of individuals who watch fireworks on television on July 4 if a random sample of 1000 Americans is selected.

Source: USA Snapshot, *USA TODAY.*

**20.Alternate Sources of Fuel** Eighty-five percent of Americans favor spending government money to develop alternative sources of fuel for automobiles. For a random sample of 120 Americans, find the mean, variance, and standard deviation for the number who favor government spending for alternative fuels.

Source: www.pollingreport.com

**21.Survey on Bathing Pets** A survey found that 25% of pet owners had their pets bathed professionally rather than do it themselves. If 18 pet owners are randomly selected, find the probability that exactly 5 people have their pets bathed professionally.

Source: USA Snapshot, *USA TODAY.*

**22.Survey on Answering Machine Ownership** In a survey, 63% of Americans said they own an answering machine. If 14 Americans are selected at random, find the probability that exactly 9 own an answering machine.

Source: USA Snapshot, *USA TODAY.*

**23.Poverty and the Federal Government** One out of every three Americans believes that the U.S. government should take “primary responsibility” for eliminating poverty in the United States. If 10 Americans are selected, find the probability that at most 3 will believe that the U.S. government should take primary responsibility for eliminating poverty.

Source: *Harper’s Index.*

**24.Internet Purchases** Thirty-two percent of adult Internet users have purchased products or services online. For a random sample of 200 adult Internet users, find the mean, variance, and standard deviation for the number who have purchased goods or services online.

Source: www.infoplease.com

**25.Survey on Internet Awareness** In a survey, 58% of American adults said they had never heard of the Internet. If 20 American adults are selected at random, find the probability that exactly 12 will say they have never heard of the Internet.

Source: *Harper’s Index.*

**26.Job Elimination** In the past year, 13% of businesses have eliminated jobs. If 5 businesses are selected at random, find the probability that at least 3 have eliminated jobs during the last year.

Source: *USA TODAY.*

**27.Survey of High School Seniors** Of graduating high school seniors, 14% said that their generation will be remembered for their social concerns. If 7 graduating seniors are selected at random, find the probability that either 2 or 3 will agree with that statement.

Source: *USA TODAY.*

**28.**Is this a binomial distribution? Explain.

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**Extending the Concepts**

**29.Children in a Family** The graph shown here represents the probability distribution for the number of girls in a family of three children. From this graph, construct a probability distribution

**30.**Construct a binomial distribution graph for the number of defective computer chips in a lot of 4 if *p* = 0.3.

**Technology Step by Step**

**MINITAB**

**Step by Step**

**The Binomial Distribution**

**Calculate a Binomial Probability**

From Example 5–19, it is known that 5% of the population is afraid of being alone at night. If a random sample of 20 Americans is selected, what is the probability that exactly 5 of them are afraid?

*n* = 20 *p* = 0.05 (5%) and *X* = 5 (5 out of 20)

No data need to be entered in the worksheet.

**1.**Select **Calc>Probability Distributions>Binomial.**

**2.**Click the option for Probability.

**3.**Click in the text box for Number of trials:.

**4.**Type in **20,** then Tab to Probability of success, then type **.05.**

**5.**Click the option for Input constant, then type in **5.** Leave the text box for Optional storage empty. If the name of a constant such as K1 is entered here, the results are stored but not displayed in the session window.

**6.**Click [OK]. The results are visible in the session window.

**Probability Density Function**

Binomial with n = 20 and p = 0.05

x f(x)

5 0.0022446

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**Construct a Binomial Distribution**

These instructions will use *n* = 20 and *p* = 0.05.

**1.**Select **Calc>Make Patterned Data>Simple Set of Numbers.**

**2.**You must enter three items:

a)Enter **X** in the box for Store patterned data in:. MINITAB will use the first empty column of the active worksheet and name it X.

b)Press Tab. Enter the value of 0 for the first value. Press Tab.

c)Enter **20** for the last value. This value should be *n.* In steps of:, the value should be 1.

**3.**Click [OK].

**4.**Select **Calc>Probability Distributions>Binomial.**

**5.**In the dialog box you must enter five items.

a)Click the button for Probability.

b)In the box for Number of trials enter **20.**

c)Enter **.05** in the Probability of success.

d)Check the button for Input columns, then type the column name, **X,** in the text box.

e)Click in the box for Optional storage, then type **Px.**

**6.**Click [OK]. The first available column will be named Px, and the calculated probabilities will be stored in it.

**7.**To view the completed table, click the worksheet icon on the toolbar.

**Graph a Binomial Distribution**

The table must be available in the worksheet.

**1.**Select **Graph>Scatterplot,** then Simple.

a)Double-click on C2 Px for the Y variable and C1 X for the X variable.

b)Click [Data view], then Project lines, then [OK]. Deselect any other type of display that may be selected in this list.

c)Click on [Labels], then Title/Footnotes.

d)Type an appropriate title, such as **Binomial Distribution n = 20, p = .05.**

e)Press Tab to the Subtitle 1, then type in Your Name.

f)Optional: Click [Scales] then [Gridlines] then check the box for Y major ticks.

g)Click [OK] twice.

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The graph will be displayed in a window. Right-click the control box to save, print, or close the graph.

**TI-83 Plus or TI-84 Plus**

**Step by Step**

**Binomial Random Variables**

To find the probability for a binomial variable:

Press **2nd [DISTR]** then 0 for binomial pdf( (Note: On the TI-84 Plus Use A)

The form is binompdf(*n,p,X*).

Example: *n* = 20, *X* = 5, *p* = .05. (Example 5–19*a* from the text) binompdf(20,.05,5)

Example: *n* = 20, *X* = 0, 1, 2, 3, *p* = .05. (Example 5–19*b* from the text) binompdf(20,.05,{0,1,2,3})

The calculator will display the probabilities in a list. Use the arrow keys to view entire display.

To find the cumulative probability for a binomial random variable:

Press **2nd [DISTR]** then **A (ALPHA MATH)** for binomcdf( (Note: On the TI-84 Plus Use B) The form is binomcdf(*n,p,X*). This will calculate the cumulative probability for values from 0 to *X*.

Example: *n* = 20, *X* = 0, 1, 2, 3, *p* = .05 (Example 5–19*b* from the text) binomcdf(20,.05,3)

To construct a binomial probability table:

**1.**Enter the *X* values 0 through *n* into L1.

**2.**Move the cursor to the top of the L2 column so that L2 is highlighted.

**3.**Type the command binompdf(*n,p,*L1), then press **ENTER.**

Example: *n* = 20, *p* = .05 (Example 5–19 from the text)

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**Excel**

**Step by Step**

**Creating a Binomial Distribution and Graph**

These instructions will demonstrate how Excel can be used to construct a binomial distribution table for *n* = 20 and *p* = 0.35.

**1.**Type **X** for the binomial variable label in cell A1 of an Excel worksheet.

**2.**Type **P(X)** for the corresponding probabilities in cell B1.

**3.**Enter the integers from 0 to 20 in column A starting at cell A2. Select the Data tab from the toolbar. Then select Data Analysis. Under Analysis Tools, select Random Number Generation and click [OK].

**4.**In the Random Number Generation dialog box, enter the following:

a)Number of Variables: **1**

b)Distribution: Patterned

c)Parameters: From **0** to **20** in steps of **1,** repeating each number: **1** times and repeating each sequence **1** times

d)Output range: **A2:A21**

**5.**Then click [OK].

**6.**To determine the probability corresponding to the first value of the binomial random variable, select cell B2 and type: **=BINOMDIST(0,20,.35,FALSE).** This will give the probability of obtaining 0 successes in 20 trials of a binomial experiment for which the probability of success is 0.35.

**7.**Repeat step **6,** changing the first parameter, for each of the values of the random variable from column A.

*Note:* If you wish to obtain the cumulative probabilities for each of the values in column A, you can type: **=BINOMDIST(0,20,.35,TRUE)** and repeat for each of the values in column A.

To create the graph:

**1.**Select the Insert tab from the toolbar and the Column Chart.

**2.**Select the Clustered Column (the first column chart under the 2-D Column selections).

**3.**You will need to edit the data for the chart.

a)Right-click the mouse on any location of the chart. Click the Select Data option. The Select Data Source dialog box will appear.

b)Click X in the Legend Entries box and click Remove.

c)Click the Edit button under Horizontal Axis Labels to insert a range for the variable X.

d)When the Axis Labels box appears, highlight cells A2 to A21 on the worksheet, then click [OK].

**4.**To change the title of the chart:

a)Left-click once on the current title.

b)Type a new title for the chart, for example, Binomial Distribution (20, .35, .65).

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**Summary**

Many variables have special probability distributions. This chapter presented several of the most common probability distributions, including the binomial distribution, the multinomial distribution, the Poisson distribution, and the hypergeometric distribution.

The binomial distribution is used when there are only two outcomes for an experiment, there are a fixed number of trials, the probability is the same for each trial, and the outcomes are independent of one another.

A probability distribution can be graphed, and the mean, variance, and standard deviation can be found. The mathematical expectation can also be calculated for a probability distribution. Expectation is used in insurance and games of chance.

**Important Terms**

discrete probability distribution

**Important Formulas**

Formula for the mean of a probability distribution:

*µ* = Σ*X* · *P*(*X*)

Formulas for the variance and standard deviation of a probability distribution:

Formula for expected value:

*E*(*X*) = Σ*X* · *P*(*X*)

Binomial probability formula:

Formula for the mean of the binomial distribution:

*µ* = *n* · *p*

Formulas for the variance and standard deviation of the binomial distribution:

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**Review Exercises**

**For Exercises 1 through 3, determine whether the distribution represents a probability distribution. If it does not, state why.**

**2.**

**4.Emergency Calls** The number of emergency calls a local police department receives per 24-hour period is distributed as shown here. Construct a graph for the data.

**5.Credit Cards** A large retail company encourages its employees to get customers to apply for the store credit card. Below is the distribution for the number of credit card applications received per employee for an 8-hour shift.

a.What is the probability that an employee will get 2 or 3 applications during any given shift?

b.Find the mean, variance, and standard deviation for this probability distribution.

**6.Coins in a Box** A box contains 5 pennies, 3 dimes, 1 quarter, and 1 half-dollar. Construct a probability distribution and draw a graph for the data.

**7.** **Tie Purchases** At Tyler’s Tie Shop, Tyler found the probabilities that a customer will buy 0, 1, 2, 3, or 4 ties, as shown. Construct a graph for the distribution.

**8.Customers in a Bank** A bank has a drive-through service. The number of customers arriving during a 15-minute period is distributed as shown. Find the mean, variance, and standard deviation for the distribution.

**9.** **Museum Visitors** At a small community museum, the number of visitors per hour during the day has the distribution shown here. Find the mean, variance, and standard deviation for the data.

**10.Cans of Paint Purchased** During a recent paint sale at Corner Hardware, the number of cans of paint purchased was distributed as shown. Find the mean, variance, and standard deviation of the distribution.

**11.** **Inquiries Received** The number of inquiries received per day for a college catalog is distributed as shown. Find the mean, variance, and standard deviation for the data.

**12.Outdoor Regatta** A producer plans an outdoor regatta for May 3. The cost of the regatta is $8000. This includes advertising, security, printing tickets, entertainment, etc. The producer plans to make $15,000 profit if all goes well. However, if it rains, the regatta will have to be canceled. According to the weather report, the probability of rain is 0.3. Find the producer’s expected profit.

**13.** **Card Game** A game is set up as follows: All the diamonds are removed from a deck of cards, and these 13 cards are placed in a bag. The cards are mixed up, and then one card is chosen at random (and then replaced). The player wins according to the following rules.

If the ace is drawn, the player loses $20.

If a face card is drawn, the player wins $10.

If any other card (2–10) is drawn, the player wins $2.

How much should be charged to play this game in order for it to be fair?

**14.**Using Exercise 13, how much should be charged if instead of winning $2 for drawing a 2–10, the player wins the amount shown on the card in dollars?

**15.**Let *x* be a binomial random variable with *n* = 12 and *p* = 0.3. Find the following:

a. *P*(*X* = 8)

b. *P*(*X* < 5)

c. *P*(*X* ≥ 10)

d. *P*(4 < X ≤ 9)

**16.Internet Access via Cell Phone** Fourteen percent of cell phone users use their cell phones to access the Internet. In a random sample of 10 cell phone users, what is the probability that exactly 2 have used their phones to access the Internet? More than 2?

Source: www.infoplease.com

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**17.** **Drug Calculation Test** If 75% of nursing students are able to pass a drug calculation test, find the mean, variance, and standard deviation of the number of students who pass the test in a sample of 180 nursing students.

**18.Flu Shots** It has been reported that 63% of adults aged 65 and over got their flu shots last year. In a random sample of 300 adults aged 65 and over, find the mean, variance, and standard deviation for the number who got their flu shots.

Source: U.S. Center for Disease Control and Prevention.

**19.** **U.S. Police Chiefs and the Death Penalty** The chance that a U.S. police chief believes the death penalty “significantly reduces the number of homicides” is 1 in 4. If a random sample of 8 police chiefs is selected, find the probability that at most 3 believe that the death penalty significantly reduces the number of homicides.

Source: *Harper’s Index.*

**20.Household Wood Burning** *American Energy Review* reported that 27% of American households burn wood. If a random sample of 500 American households is selected, find the mean, variance, and standard deviation of the number of households that burn wood.

Source: *100% American* by Daniel Evan Weiss.

**21.** **Pizza for Breakfast** Three out of four American adults under age 35 have eaten pizza for breakfast. If a random sample of 20 adults under age 35 is selected, find the probability that exactly 16 have eaten pizza for breakfast.

Source: *Harper’s Index.*

**22.Unmarried Women** According to survey records, 75.4% of women aged 20–24 have never been married. In a random sample of 250 young women aged 20–24, find the mean, variance, and standard deviation for the number who are or who have been married.

Source: www.infoplease.com

**Statistics Today**

**Is Pooling Worthwhile?—Revisited**

In the case of the pooled sample, the probability that only one test will be needed can be determined by using the binomial distribution. The question being asked is, In a sample of 15 individuals, what is the probability that no individual will have the disease? Hence, *n* = 15, *p* = 0.05, and *X* = 0. From Table B in Appendix C, the probability is 0.463, or 46% of the time, only one test will be needed. For screening purposes, then, pooling samples in this case would save considerable time, money, and effort as opposed to testing every individual in the population.

**Chapter Quiz**

**Determine whether each statement is true or false. If the statement is false, explain why.**

**1.** The expected value of a random variable can be thought of as a long-run average.

**2.** The number of courses a student is taking this semester is an example of a continuous random variable.

**3.** When the binomial distribution is used, the outcomes must be dependent.

**4.** A binomial experiment has a fixed number of trials.

**Complete these statements with the best answer.**

**5.** Random variable values are determined by _______.

**6.** The mean for a binomial variable can be found by using the formula _______.

**7.** One requirement for a probability distribution is that the sum of all the events in the sample space must equal ______.