# Mathematics

Select the best answer.

8. What is the sum of the probabilities of all outcomes in a probability distribution?

a.0

b.

c.1

d.It cannot be determined.

9. How many outcomes are there in a binomial experiment?

a.0

b.1

c.2

d.It varies.

10. The number of trials for a binomial experiment

a.Can be infinite

b.Is unchanged

c.Is unlimited

d.Must be fixed

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For questions 11 through 14, determine if the distribution represents a probability distribution. If not, state why.

11.

12.

13.

14.

15. Calls for a Fire Company The number of fire calls the Conestoga Valley Fire Company receives per day is distributed as follows:

Construct a graph for the data.

16. Telephones per Household A study was conducted to determine the number of telephones each household has. The data are shown here.

Construct a probability distribution and draw a graph for the data.

17. CD Purchases During a recent CD sale at Matt’s Music Store, the number of CDs customers purchased was distributed as follows:

Find the mean, variance, and standard deviation of the distribution.

18. Calls for a Crisis Hot Line The number of calls received per day at a crisis hot line is distributed as follows:

Find the mean, variance, and standard deviation of the distribution.

19. Selecting a Card There are 6 playing cards placed face down in a box. They are the 4 of diamonds, the 5 of hearts, the 2 of clubs, the 10 of spades, the 3 of diamonds, and the 7 of hearts. A person selects a card. Find the expected value of the draw.

20. Selecting a Card A person selects a card from an ordinary deck of cards. If it is a black card, she wins $2. If it is a red card between or including 3 and 7, she wins $10. If it is a red face card, she wins $25; and if it is a black jack, she wins an extra $100. Find the expectation of the game.

21. Carpooling If 40% of all commuters ride to work in carpools, find the probability that if 8 workers are selected, 5 will ride in carpools.

22.Employed Women If 60% of all women are employed outside the home, find the probability that in a sample of 20 women,

a. Exactly 15 are employed

b. At least 10 are employed

c. At most 5 are not employed outside the home

23. Driver’s Exam If 80% of the applicants are able to pass a driver’s proficiency road test, find the mean, variance, and standard deviation of the number of people who pass the test in a sample of 300 applicants.

24. Meeting Attendance A history class has 75 members. If there is a 12% absentee rate per class meeting, find the mean, variance, and standard deviation of the number of students who will be absent from each class.

Critical Thinking Challenges

1.Lottery Numbers Pennsylvania has a lottery entitled “Big 4.” To win, a player must correctly match four digits from a daily lottery in which four digits are selected. Find the probability of winning.

2.Lottery Numbers In the Big 4 lottery, for a bet of $100, the payoff is $5000. What is the expected value of winning? Is it worth it?

3.Lottery Numbers If you played the same four-digit number every day (or any four-digit number for that matter) in the Big 4, how often (in years) would you win, assuming you have average luck?

4.Chuck-a-Luck In the game Chuck-a-Luck, three dice are rolled. A player bets a certain amount (say $1.00) on a number from 1 to 6. If the number appears on 1 die, the person wins $1.00. If it appears on 2 dice, the person wins $2.00, and if it appears on all 3 dice, the person wins $3.00. What are the chances of winning $1.00? $2.00? $3.00?

5.Chuck-a-Luck What is the expected value of the game of Chuck-a-Luck if a player bets $1.00 on one number?

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Data Projects

1.Business and Finance Assume that a life insurance company would like to make a profit of $250 on a $100,000 policy sold to a person whose probability of surviving the year is 0.9985. What premium should the company charge the customer? If the company would like to make a $250 profit on a $100,000 policy at a premium of $500, what is the lowest life expectancy it should accept for a customer?

2.Sports and Leisure Baseball, hockey, and basketball all use a seven-game series to determine their championship. Find the probability that with two evenly matched teams a champion will be found in 4 games. Repeat for 5, 6, and 7 games. Look at the historical results for the three sports. How do the actual results compare to the theoretical?

3.Technology Use your most recent itemized phone bill for the data in this problem. Assume that incoming and outgoing calls are equal in the population (why is this a reasonable assumption?). This means assume p = 0.5. For the number of calls you made last month, what would be the mean number of outgoing calls in a random selection of calls? Also, compute the standard deviation. Was the number of outgoing calls you made an unusual amount given the above? In a selection of 12 calls, what is the probability that less than 3 were outgoing?

4.Health and Wellness Use Red Cross data to determine the percentage of the population with an Rh factor that is positive (A+, B+, AB+ , or O+ blood types). Use that value for p. How many students in your class have a positive Rh factor? Is this an unusual amount?

5.Politics and Economics Find out what percentage of citizens in your state is registered to vote. Assuming that this is a binomial variable, what would be the mean number of registered voters in a random group of citizens with a sample size equal to the number of students in your class? Also determine the standard deviation. How many students in your class are registered to vote? Is this an unusual number, given the above?

6.Your Class Have each student in class toss 4 coins on her or his desk, and note how many heads are showing. Create a frequency table displaying the results. Compare the frequency table to the theoretical probability distribution for the outcome when 4 coins are tossed. Find the mean for the frequency table. How does it compare with the mean for the probability distribution?

Answers to Applying the Concepts

Section 5–1 Dropping College Courses

1.The random variable under study is the reason for dropping a college course.

2.There were a total of 144 people in the study.

3.The complete table is as follows:

Reason for dropping a college course

Frequency

Percentage

Too difficult

45

31.25

Illness

40

27.78

Change in work schedule

20

13.89

Change of major

14

9.72

Family-related problems

9

6.25

Money

7

4.86

Miscellaneous

6

4.17

No meaningful reason

3

2.08

4.The probability that a student will drop a class because of illness is about 28%. The probability that a student will drop a class because of money is about 5%. The probability that a student will drop a class because of a change of major is about 10%.

5.The information is not itself a probability distribution, but it can be used as one.

6.The categories are not necessarily mutually exclusive, but we treated them as such in computing the probabilities.

7.The categories are not independent.

8.The categories are exhaustive.

9.Since all the probabilities are between 0 and 1, inclusive, and the probabilities sum to 1, the requirements for a discrete probability distribution are met.

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Section 5–2 Expected Value

1.The expected value is the mean in a discrete probability distribution.

2.We would expect variation from the expected value of 3.

3.Answers will vary. One possible answer is that pregnant mothers in that area might be overly concerned upon hearing that the number of cases of kidney problems in newborns was nearly 4 times what was usually expected. Other mothers (particularly those who had taken a statistics course!) might ask for more information about the claim.

4.Answers will vary. One possible answer is that it does seem unlikely to have 11 newborns with kidney problems when we expect only 3 newborns to have kidney problems.

5.The public might better be informed by percentages or rates (e.g., rate per 1000 newborns).

6.The increase of 8 babies born with kidney problems represents a 0.32% increase (less than %).

7.Answers will vary. One possible answer is that the percentage increase does not seem to be something to be overly concerned about.

Section 5–3 Unsanitary Restaurants

1.The probability of eating at 3 restaurants with unsanitary conditions out of the 10 restaurants is 0.18651.

2.The probability of eating at 4 or 5 restaurants with unsanitary conditions out of the 10 restaurants is 0.24623 + 0.22291 = 0.46914.

3.To find this probability, you could add the probabilities for eating at 1, 2, … , 10 unsanitary restaurants. An easier way to compute the probability is to subtract the probability of eating at no unsanitary restaurants from 1 (using the complement rule).

4.The highest probability for this distribution is 4, but the expected number of unsanitary restaurants that you would eat at is 10 • = 4.3.

5.The standard deviation for this distribution is = 1.56.

6.This is a binomial distribution. We have two possible outcomes: “success” is eating in an unsanitary restaurant; “failure” is eating in a sanitary restaurant. The probability that one restaurant is unsanitary is independent of the probability that any other restaurant is unsanitary. The probability that a restaurant is unsanitary remains constant at . And we are looking at the number of unsanitary restaurants that we eat at out of 10 “trials.”

7.The likelihood of success will vary from situation to situation. Just because we have two possible outcomes, this does not mean that each outcome occurs with probability 0.50.