Question 1

Sixty-five percent of men consider themselves knowledgeable football fans. If 10 men are randomly selected, find the probability that exactly six of them will consider themselves knowledgeable fans.

Accepted Answer

This is a binomial probability problem where n = 10 (number of trials), k = 6 (number of successes), and p = 0.65 (probability of success on a single trial). The binomial probability formula is P(X = k) = (nCk) * p^k * (1-p)^(n-k), where nCk is the combination of n items taken k at a time. Plugging in the values: P(X = 6) = (10C6) * (0.65)^6 * (0.35)^4, which calculates to approximately 0.238.

Question 2

Twenty-six percent of people in the United States with Internet access go online to get news. A random
sample of five Americans with Internet access is selected and asked if they get the news online. Identify the
values of n, p, and q, and list the possible values of the random variable x.

Accepted Answer

The answer of Twenty-six percent of people in the United...

Question 3

A company ships computer components in boxes that contain 10 items. Assume that the probability of a
defective computer component is 0.1. Use the geometric mean to find the mean number of defective parts.
Interpret the results.

Accepted Answer

The answer of A company ships computer components in boxes...

Question 4

An airline has a policy of booking as many as 150 persons on a plane that seats 140. Past studies indicate that
only 85% of booked passengers show up for their flight. Find the probability that if the airline books 150
persons for a 140-seat plane, not enough seats will be available.

Accepted Answer

The answer of An airline has a policy of booking...

Question 5

Basketball player Chauncey Billups of the Detroit Pistons makes free throw shots 88% of the time. Find the probability that he misses his first shot and makes the second.

Accepted Answer

The probability of missing the first shot is 12% (100% - 88%), and the probability of making the second shot is 88%. Multiplying these probabilities gives 0.12 * 0.88 = 0.1056.

Question 6

A statistics professor finds that when he schedules an office hour at the 10:30 a.m. time slot, an average of three students arrives. Use the Poisson distribution to find the probability that in a randomly selected office hour no
Students will arrive.

Accepted Answer

The probability that no students will arrive during a randomly selected office hour, given an average arrival rate ($\lambda$) of 3 students, can be calculated using the Poisson distribution formula for $k = 0$ events: $P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$. Substituting $\lambda = 3$ and $k = 0$, we get $P(X=0) = \frac{e^{-3} 3^0}{0!} = e^{-3} \approx 0.0498$.

Question 7

A statistics professor finds that when he schedules an office hour at the 10:30 a.m. time slot, an average of three students arrive. Use the Poisson distribution to find the probability that in a randomly selected office hour in
The 10:30 a.m. time slot exactly two students will arrive.

Accepted Answer

The probability of exactly k events in a Poisson distribution is given by $P(k; \lambda) = \frac{e^{-\lambda} \lambda^k}{k!}$, where $\lambda$ is the average rate (3 students in this case) and $k$ is the number of events (2 students). Plugging in the values: $P(2; 3) = \frac{e^{-3} 3^2}{2!} = 0.2241$.

Question 8

Fifty-seven percent of families say that their children have an influence on their vacation plans. Consider a
sample of eight families who are asked if their children influence their vacation plans. Identify the values of n,
p, and q, and list the possible values of the random variable x.

Accepted Answer

The answer of Fifty-seven percent of families say that their...

Question 9

A company ships computer components in boxes that contain 90 items. Assume that the probability of a
defective computer component is 0.21. Find the probability that the first defect is found in the seventh
component tested. Round your answer to four decimal places.

Accepted Answer

The answer of A company ships computer components in boxes...

Question 10

A mail-order company receives an average of five orders per 500 solicitations. If it sends out 100 advertisements, find the probability of receiving at least two orders. Use the Poisson distribution.

Accepted Answer

The Poisson distribution is given by:P(X = k) = (e^-λ * λ^k) / k!where λ is the average number of events per unit time.In this case, λ = 5 / 500 = 0.01.The probability of receiving at least two orders is:P(X >= 2) = 1 - P(X = 0) - P(X = 1)= 1 - (e^-0.01 * 0.01^0) / 0! - (e^-0.01 * 0.01^1) / 1!= 1 - 0.9900 - 0.0099= 0.0001Therefore, the probability of receiving at least two orders is 0.0001, which is closest to choice A) 0.2642.

Question 11

A sales firm receives an average of three calls per hour on its toll-free number. For any given hour, find the probability that it will receive at least three calls. Use the Poisson distribution.

Accepted Answer

The probability of receiving at least three calls in an hour is given by the complement of the probability of receiving less than three calls. The Poisson distribution is given by:P(X = k) = (e^-λ * λ^k) / k!where λ is the average number of calls per hour. In this case, λ = 3.The probability of receiving less than three calls is:P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)= (e^-3 * 3^0) / 0! + (e^-3 * 3^1) / 1! + (e^-3 * 3^2) / 2!= 0.0498 + 0.1494 + 0.2240= 0.4232Therefore, the probability of receiving at least three calls is:P(X >= 3) = 1 - P(X < 3)= 1 - 0.4232= 0.5768

Question 12

A book contains 500 pages. If there are 200 typing errors randomly distributed throughout the book, use the Poisson distribution to determine the probability that a page contains exactly two errors.

Accepted Answer

We are given that the Poisson distribution applies and we can use the formula:

P(X = k) = (e^-λ * λ^k) / k!

where λ is the mean and equal to the total number of errors divided by the number of pages:

λ = 200 / 500 = 0.4

We want to find the probability that a page contains exactly two errors, so we set k = 2:

P(X = 2) = (e^-0.4 * 0.4^2) / 2! = 0.0536

Therefore, the answer is A) 0.0536. We can use the Poisson probability function given by:

P(X = k) = e^(-λ) * λ^k / k!

where X is the random variable (number of errors on a given page), k is the specific value of X we are interested in (2), and λ is the expected value of X (the average number of errors per page).

To find λ, we can divide the total number of errors by the total number of pages:

λ = 200 / 500 = 0.4

Now we can plug in these values:

P(X = 2) = e^(-0.4) * 0.4^2 / 2! = 0.0536

Therefore, the probability that a page contains exactly two errors is 0.0536 or approximately 5.36%.

Question 13

Seventy percent of adults ages 50 and older want to reach their 100th birthday. You randomly select nine adults
ages 50 and older and ask them whether they want to reach their 100th birthday. The random variable
represents the number of adults ages 50 and older who want to reach their 100th birthday. Graph the binomial
distribution using a histogram and describe its shape.

Accepted Answer

The answer of Seventy percent of adults ages 50 and...

Question 14

A sales firm receives an average of four calls per hour on its toll-free number. For any given hour, find the probability that it will receive exactly nine calls. Use the Poisson distribution.

Accepted Answer

This problem involves using the Poisson distribution, which gives the probability of a certain number of events occurring in a fixed interval of time, given the average rate of events occurring. In this case, the average rate is four calls per hour, and we want to find the probability of nine calls in one hour. Using the Poisson formula, P(X=9) = (e^-4)*(4^9)/9! = 0.0132. Therefore, the correct answer is choice A. This problem involves finding the probability of a rare event (receiving exactly 9 calls in an hour) using the Poisson distribution. The Poisson distribution models rare events that occur randomly over time or space, such as accidents, defects, or phone calls. The Poisson distribution has one parameter λ (lambda), which represents the average rate or intensity of the events. This parameter is also equal to the mean and variance of the distribution. Therefore, if the average rate is 4 calls per hour, we can set λ=4 and use the Poisson formula:

P(X=9) = (e^(-λ) * λ^9) / 9! = (e^(-4) * 4^9) / 362880 ≈ 0.0132

where X is the random variable representing the number of calls in an hour, and e is the base of natural logarithms. The probability of getting exactly 9 calls in an hour is very low (less than 2%), but not impossible. Therefore, the best choice is A) 0.0132. The other choices are either too high (C) or too low (B, D) to be plausible probabilities.

Question 15

A car towing service company averages two calls per hour. Use the Poisson distribution to determine the probability that in a randomly selected hour the number of calls is six.

Accepted Answer

The Poisson distribution is given by:P(X = k) = (e^-λ * λ^k) / k!where:λ is the average number of calls per hourk is the number of calls in a randomly selected hourIn this case, λ = 2 and k = 6. So, the probability that in a randomly selected hour the number of calls is six is:P(X = 6) = (e^-2 * 2^6) / 6! = 0.0120

Question 16

Thirty-eight percent of people in the United States have type O+ blood. You randomly select 30 Americans
and ask them if their blood type is O+. Identify the values of n, p, and q, and list the possible values of the
random variable x.

Accepted Answer

The answer of Thirty-eight percent of people in the United...

Question 17

Thirty percent of voters think that Congress should help write standards for school food. You randomly select
eight voters and ask them whether Congress should help write standards for school food. The random variable
represents the number of voters who think that Congress should help write standards for school food. Graph
the binomial distribution using a histogram and describe its shape.

Accepted Answer

The answer of Thirty percent of voters think that Congress...

Question 18

A company ships computer components in boxes that contain 10 items. Assume that the probability of a defective computer component is 0.1. Use the geometric variance to find the variance of defective parts.

Accepted Answer

The variance of a geometric distribution is given by $ \frac{1-p}{p^2} $, where $ p $ is the probability of success (in this case, finding a defective part). Given $ p = 0.1 $, the variance is $ \frac{1-0.1}{0.1^2} = \frac{0.9}{0.01} = 90 $.

Question 19

A local fire station receives an average of 0.55 rescue calls per day. Use the Poisson distribution to find the probability that on a randomly selected day, the fire station will receive fewer than two calls.

Accepted Answer

The Poisson distribution formula for the probability of observing k events in a fixed interval is P(k; λ) = (e^(-λ) * λ^k) / k!, where λ is the average number of events (0.55 in this case) and k is the number of occurrences (here, we're interested in fewer than 2 calls, so we calculate for k=0 and k=1). For k=0: P(0; 0.55) = (e^(-0.55) * 0.55^0) / 0! = e^(-0.55). For k=1: P(1; 0.55) = (e^(-0.55) * 0.55^1) / 1! = 0.55 * e^(-0.55). Adding these probabilities gives the total probability of receiving fewer than two calls. The calculation matches closest to option A, indicating a high probability (0.894) of receiving fewer than two calls.

Question 20

You observe the gender of the next 100 babies born at a local hospital. You count the number of girls born.
Identify the values of n, p, and q, and list the possible values of the random variable x.

Accepted Answer

The answer of You observe the gender of the next...

Quiz 4: Discrete Probability Distributions

Sixty-five percent of men consider themselves knowledgeable football fans. If 10 men are randomly selected, find the probability that exactly six of them will consider themselves knowledgeable fans.

Twenty-six percent of people in the United States with Internet access go online to get news. A random sample of five Americans with Internet access is selected and asked if they get the news online. Identify the values of n, p, and q, and list the possible values of the random variable x.

A company ships computer components in boxes that contain 10 items. Assume that the probability of a defective computer component is 0.1. Use the geometric mean to find the mean number of defective parts. Interpret the results.

An airline has a policy of booking as many as 150 persons on a plane that seats 140. Past studies indicate that only 85% of booked passengers show up for their flight. Find the probability that if the airline books 150 persons for a 140-seat plane, not enough seats will be available.

Basketball player Chauncey Billups of the Detroit Pistons makes free throw shots 88% of the time. Find the probability that he misses his first shot and makes the second.

A statistics professor finds that when he schedules an office hour at the 10:30 a.m. time slot, an average of three students arrives. Use the Poisson distribution to find the probability that in a randomly selected office hour no Students will arrive.

A statistics professor finds that when he schedules an office hour at the 10:30 a.m. time slot, an average of three students arrive. Use the Poisson distribution to find the probability that in a randomly selected office hour in The 10:30 a.m. time slot exactly two students will arrive.

Fifty-seven percent of families say that their children have an influence on their vacation plans. Consider a sample of eight families who are asked if their children influence their vacation plans. Identify the values of n, p, and q, and list the possible values of the random variable x.

A company ships computer components in boxes that contain 90 items. Assume that the probability of a defective computer component is 0.21. Find the probability that the first defect is found in the seventh component tested. Round your answer to four decimal places.