# Probability

According to the "January theory," if the stock market is up for the month of January, it will be up for the year. If it is down in January, it will be down for the year. According to an article in The Wall Street Journal, this theory held for 28 out of the last 34 years. Suppose there is no truth to this theory; that is, the probability it is either up or down is 0.5.

What is the probability this could occur by chance? (Round your answer to 6 decimal places.)

Probability

The United States Postal Service reports 95 percent of first class mail within the same city is delivered within two days of the time of mailing. Six letters are randomly sent to different locations.

(a) What is the probability that all six arrive within two days? (Round your answer to 4 decimal places.)

Probability

(b) What is the probability that exactly five arrive within two days? (Round your answer to 4 decimal places.)

Probability

(c) Find the mean number of letters that will arrive within two days. (Round your answer to 1 decimal place.)

Number of letters

(d-1) Compute the variance of the number that will arrive within two days. (Round your answer to 3 decimal places.)

Variance

(d-2) Compute the standard deviation of the number that will arrive within two days. (Round your answer to 4 decimal places.)

Standard Deviation

A CD contains 10 songs; 6 are classical and 4 are rock and roll.

In a sample of 3 songs, what is the probability that exactly 2 are classical? Assume the samples are drawn without replacement. (Round your answer to 2 decimal places.)

Probability

An internal study by the Technology Services department at Lahey Electronics revealed company employees receive an average of 6.4 emails per hour. Assume the arrival of these emails is approximated by the Poisson distribution.

(a) What is the probability Linda Lahey, company president, received exactly 3 email between 4 P.M. and 5 P.M. yesterday? (Round your answer to 4 decimal places.)

Probability

(b) What is the probability she received 8 or more emails during the same period? (Round your answer to 4 decimal places.)

Probability

(c) What is the probability she received four or less email during the period? (Round your answer to 4 decimal places.)

Probability

Suppose the Internal Revenue Service is studying the category of charitable contributions. A sample of 34 returns is selected from young couples between the ages of 20 and 35 who had an adjusted gross income of more than $100,000. Of these 34 returns 8 had charitable contributions of more than $1,000. Suppose 7 of these returns are selected for a comprehensive audit.

(a) You should use the hypergeometric distribution because

(b) What is the probability exactly one of the seven audited had a charitable deduction of more than $1,000? (Round your answer to 4 decimal places.)

Probability

(c) What is the probability at least one of the audited returns had a charitable contribution of more than $1,000? (Round your answer to 3 decimal places.)

Probability

Suppose 1.5 percent of the antennas on new Nokia cell phones are defective. For a random sample of 200 antennas, find the probability that: (Use the poisson approximation to the binomial.)

(a) None of the antennas are defective. (Round your answer to 4 decimal places.)

Probability

(b) Three or more of the antennas are defective. (Round your answer to 4 decimal places.)

Probability

The game called Lotto sponsored by the Louisiana Lottery Commission pays its largest prize when a contestant matches all 6 of the 35 possible numbers. Assume there are 35 ping-pong balls each with a single number between 1 and 35. Any number appears only once, and the winning balls are selected without replacement.

(a) The commission reports that the probability of matching all the numbers are 1 in 1,623,160. What is this in terms of probability? (Round your answer to 8 decimal places.)

Probability

(b) Find the probability, again using the hypergeometric formula, for matching 4 of the 6 winning numbers. (Round your answer to 8 decimal places.)

Probability

(c) Find the probability of matching 5 of the 6 winning numbers. (Round your answer to 8 decimal places.)

Probability

In a binomial distribution and . Find the probabilities of the following events. (Round your answers to 4 decimal places.)

(a)

Probability

(b)

Probability

(c)

Probability

A manufacturer of window frames knows from long experience that 5 percent of the production will have some type of minor defect that will require an adjustment. What is the probability that in a sample of 20 window frames:

(a) None will need adjustment? (Round your answer to 3 decimal places.)

Probability

(b) At least one will need adjustment? (Round your answer to 3 decimal places.)

Probability

(c) More than two will need adjustment? (Round your answer to 3 decimal places.)

Probability

Three tables listed below show random variables and their probabilities. However, only one of these is actually a probability distribution.

(a) Which is it?

A B C

x P(x) x P(x) x P(x)

5 .2 5 .2 5 .3

10 .3 10 .2 10 .2

15 .1 15 .1 15 .2

20 .7 20 -.5 20 .3

(b) Using the correct probability distribution, find the probability that x is: (Round your answers to 1 decimal place.)

(a) P(Exactly 15) =

(b) P(No more than 10) =

(c) P(More than 5) =

c) Compute the mean, variance, and standard deviation of this distribution. (Round your answers to 2 decimal places.)

(a) Mean µ

(b) Variance σ2

(c) Standard deviation σ

The director of admissions at Kinzua University in Nova Scotia estimated the distribution of student admissions for the fall semester on the basis of past experience.

Admissions Probability

1,030 0.2

1,380 0.2

1,580 0.6

________________________________________

(1) What is the expected number of admissions for the fall semester? (Round your answers to the nearest whole number.)

Expected number of admissions

(2) Compute the variance and the standard deviation of the number of admissions. (Round your answers to 2 decimal places.)

Variance

Standard deviation

In a recent study 90 percent of the homes in the United States were found to have largescreen TVs. In a sample of nine homes, calculate the probabilities.

(a) All nine have large-screen TVs. (Round your answer to 3 decimal places.)

Probability

(b) Less than five have large-screen TVs. (Round your answer to 3 decimal places.)

Probability

(c) More than five have large-screen TVs. (Round your answer to 3 decimal places.)

Probability

(d) At least seven homes have large-screen TVs. (Round your answer to 3 decimal places.)

Probability

Compute the mean and variance of the following discrete probability distribution.

X P(x)

2 .5

8 .3

10 .2

________________________________________

(Round your answers to 2 decimal places.)

(1)

μ =

(2)

σ ² =

The Downtown Parking Authority of Tampa, Florida, reported the following information for a sample of 234 customers on the number of hours cars are parked and the amount they are charged.

Number of Hours Frequency Amount Charged

1 17 $3

2 35 8

3 49 10

4 44 16

5 33 22

6 15 24

7 9 28

8 32 30

234

________________________________________

(a) Convert the information on the number of hours parked to a probability distribution. (Round your answers to 3 decimal places.)

Hours Probability

1

2

3

4

5

6

7

8

________________________________________

(b-1) Find the mean and the standard deviation of the number of hours parked. (Round your intermediate values and final answers to 3 decimal places.)

Mean

Standard deviation

(b-2) How long is a typical customer parked? (Round your answer to 2 decimal places.)

The typical customer is parked for hours

(c) Find the mean and the standard deviation of the amount charged. (Round your intermediate values and final answers to 3 decimal places.)

Mean

Standard deviation

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Worksheet Difficulty: Medium Learning Objective: 06-3

Which of these variables are discrete and which are continuous random variables?

(a) The number of new accounts established by a salesperson in a year.

(b) The time between customer arrivals to a bank ATM.

(c) The number of customers in Big Nicks barber shop.

(d) The amount of fuel in your cars gas tank.

(e) The number of minorities on a jury.

(f) The outside temperature today.

It is reported that 32 percent of American households use a cell phone exclusively for their telephone service. In a sample of fourteen households, find the probability that:

(a) None use a cell phone as their exclusive service. (Round your answer to 4 decimal places.)

Probability

(b) At least one uses the cell exclusively. (Round your answer to 4 decimal places.)

Probability

(c) At least eight use the cell phone. (Round your answer to 4 decimal places.)

Probability

The Downtown Parking Authority of Tampa, Florida, reported the following information for a sample of 250 customers on the number of hours cars are parked and the amount they are charged.

Number of Hours Frequency Amount Charged

1 20 $3.00

2 38 6.00

3 53 9.00

4 45 12.00

5 40 14.00

6 13 16.00

7 5 18.00

8 36 20.00

250

________________________________________

(a-1) Convert the information on the number of hours parked to a probability distribution. (Round your answers to 3 decimal places.)

Hours Probability

1

2

3

4

5

6

7

8

(a-2) Is this a discrete or a continuous probability distribution?

(b-1) Find the mean and the standard deviation of the number of hours parked. (Round your answer to 2 decimal places.)

Mean

Standard deviation

(b-2) How long is a typical customer parked? (Round your answer to 2 decimal places.)

The typical customer is parked for

(c) Find the mean and the standard deviation of the amount charged. (Round your answer to 2 decimal places.)

Mean

Standard deviation

It is asserted that 60 percent of the cars approaching an individual toll booth in New Jersey are equipped with an E-ZPass transponder. Find the probability that in a sample of five cars:

(a) All five will have the transponder. (Round your answer to 4 decimal places.)

Probability

(b) At least two will have the transponder. (Round your answer to 4 decimal places.)

Probability

(c) None will have a transponder. (Round your answer to 6 decimal places.)

Probability

Listed below is the population by state for the 15 states with the largest population. Also included is whether that state's border touches the Gulf of Mexico, the Atlantic Ocean, or the Pacific Ocean (coastline).

Rank State Population Coastline

1 California 35,893,799 Yes

2 Texas 22,490,022 Yes

3 New York 19,227,088 Yes

4 Florida 17,397,161 Yes

5 Illinois 12,713,634 No

6 Pennsylvania 12,406,292 No

7 Ohio 11,459,011 No

8 Michigan 10,112,620 No

9 Georgia 8,829,383 Yes

10 North Carolina 8,541,221 Yes

11 New Jersey 8,698,879 Yes

12 Virginia 7,459,827 Yes

13 Washington 6,203,788 Yes

14 Massachusetts 6,416,505 Yes

15 Indiana 6,237,569 No

________________________________________

Suppose four states are selected at random. Calculate the probability for the following:

(a) None of the states selected have a population of more than 12000000 (Round your answer to 3 decimal places.)

Probability

(b) Exactly one of the selected states has a population of more than 12000000 (Round your answer to 2 decimal places.)

Probability

(c) At least one of the selected states has a population of more than 12000000 (Round your answer to 3 decimal places.)

Probability

See attached file.

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#### Solution Summary

The solution provides step by step method for the calculation of binomial probabilities. Working are included in the excel sheet.