Question 1

Compute Probabilities of a Poisson Random Variable
-The local police department receives an average of two calls per hour. Use the Poisson distribution to determine the probability that in a randomly selected hour the number of calls is five.
A) 0.0361
B) 0.0018
C) 0.0282
D) 0.0014

Accepted Answer

The probability of observing $k$ events in a fixed interval in a Poisson distribution is given by $P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$, where $\lambda$ is the average number of events per interval. Here, $\lambda = 2$ and $k = 5$. Thus, $P(X=5) = \frac{e^{-2} 2^5}{5!} = \frac{e^{-2} 32}{120} \approx 0.0361$.

Question 2

Compute Probabilities of a Poisson Random Variable
-An online retailer receives an average of five orders per 500 hits on its website. If it gets 100 hits on its website, find the probability of receiving at least two orders. Use the Poisson distribution.
A) 0.2642
B) 0.1839
C) 0.9048
D) 0.9596

Accepted Answer

The answer of Compute Probabilities of a Poisson Random Variable
-An...

Question 3

Construct Binomial Probability Histograms
-Given that a random variable x, the number of successes, follows a Poisson process, then the number of successes in any interval is independent of the number of successes in any other interval provided the intervals
A) are disjoint.
B) overlap.
C) have at least one element in common.
D) are the same size and are independent.

Accepted Answer

The Poisson process assumes that the probability of a success in any given interval is proportional to the length of the interval, and that the number of successes in any interval follows a Poisson distribution. Therefore, the intervals must be disjoint (i.e. non-overlapping) in order for the number of successes in one interval to be independent of the number of successes in another interval. If the intervals were to overlap, then the number of successes in one interval could potentially affect the number of successes in another interval, violating the assumption of independence.

Question 4

Compute Probabilities of a Poisson Random Variable
-A local animal rescue organization receives an average of 0.55 rescue calls per hour. Use the Poisson distribution to find the probability that during a randomly selected hour, the organization will receive fewer than two calls.
A) 0.894
B) 0.106
C) 0.317
D) 0.087

Accepted Answer

Let X be the number of rescue calls received by the organization in an hour. We know that X follows a Poisson distribution with parameter λ = 0.55.
We need to find P(X < 2) = P(X = 0) + P(X = 1).
Using the Poisson probability mass function, we get:
P(X = 0) = e^-λ * λ^0 / 0! = e^-0.55 * 0.55^0 / 1 = 0.58
P(X = 1) = e^-λ * λ^1 / 1! = e^-0.55 * 0.55^1 / 1 = 0.32
Therefore, P(X < 2) = 0.58 + 0.32 = 0.894. Thus, the best choice is A.

Question 5

Compute Probabilities of a Poisson Random Variable
-A customer service 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.
A) 0.5768
B) 0.1891
C) 0.4232
D) 0.6138

Accepted Answer

The answer of Compute Probabilities of a Poisson Random Variable
-A...

Question 6

Compute Probabilities of a Poisson Random Variable
-Suppose the number of babies born during an 8-hour shift at a hospital s maternity wing follows a Poisson distribution with a mean of 7 an hour. Some people believe that the presence of a full moon increases the number of births that take place. Suppose during the presence of a full moon, County Hospital experienced eight consecutive hours with more than eight births. Based on this fact, comment on the belief that the full moon increases the number of births.
A) The belief is supported as the probability of observing this many births would be 0.0000291.
B) The belief is not supported as the probability of observing this many births is 0.271.
C) The belief is supported as the probability of observing this many births would be 0.271.
D) The belief is not supported as the probability of observing this many births is 0.0000291.

Accepted Answer

The belief is supported because the probability of observing more than eight births in an eight-hour period when the mean is 7 an hour is very low, making the observed increase during a full moon statistically significant.

Question 7

Compute Probabilities of a Poisson Random Variable
-The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 6.6. Find the probability that the next two months will both result in four accidents each occurring on this stretch of road.
A) 0.011568
B) 0.107553
C) 0.215105
D) 0.002341

Accepted Answer

The probability of observing exactly $k$ accidents in a month, when accidents follow a Poisson distribution with mean $\lambda$, is given by $P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$. Here, $\lambda = 6.6$ and $k = 4$. So, for one month, $P(X=4) = \frac{e^{-6.6} 6.6^4}{4!}$. Since the number of accidents in each month is independent, the probability of observing 4 accidents in both months is the product of the probabilities for each month: $P(X=4)^2 = \left(\frac{e^{-6.6} 6.6^4}{4!}\right)^2 \approx 0.011568$.

Question 8

Compute Probabilities of a Poisson Random Variable
-A history 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 five students will arrive.
A) 0.1008
B) 0.0137
C) 0.0519
D) 0.0070

Accepted Answer

The probability of exactly k arrivals in a Poisson distribution with mean λ is given by:P(X = k) = (e^-λ * λ^k) / k!In this case, λ = 3 and k = 5, so:P(X = 5) = (e^-3 * 3^5) / 5! = 0.1008

Question 9

Construct Binomial Probability Histograms
-Given that a random variable x, the number of successes, follows a Poisson process, then the probability of success for any two intervals of the same size
A) is the same.
B) are complementary.
C) are reciprocals.
D) none of these

Accepted Answer

In a Poisson process, the probability of success (the arrival of an event) is constant over time intervals of the same size. Therefore, any two intervals of the same size will have the same probability of success.

Question 10

Compute Probabilities of a Poisson Random Variable
-The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 8.9. Find the probability that less than three accidents will occur next month on this stretch of road.
A) 0.006752
B) 0.022777
C) 0.993248
D) 0.977223

Accepted Answer

The answer of Compute Probabilities of a Poisson Random Variable
-The...

Question 11

Construct Binomial Probability Histograms
-Draw the probability histogram and label the mean for n = 9 and p = 0.7

Accepted Answer

The answer of Construct Binomial Probability Histograms
-Draw the probability histogram...

Question 12

Construct Binomial Probability Histograms
-Given that a random variable x, the number of successes, follows a Poisson process, the probability of 2 or more successes in any sufficiently small subinterval is
A) zero.
B) one.
C) any number between zero and one.
D) none of these

Accepted Answer

In a Poisson process, the probability of more than one success in a small interval approaches zero as the interval becomes smaller. Therefore, the probability of 2 or more successes in any sufficiently small subinterval is practically zero.

Question 13

Compute Probabilities of a Poisson Random Variable
-A help desk 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 seven calls. Use the Poisson distribution.
A) 0.0596
B) 0.0030
C) 522.0448
D) 0.0087

Accepted Answer

The probability of receiving exactly seven calls in an hour, given an average rate ($\lambda$) of 4 calls per hour, can be calculated using the Poisson distribution formula: $P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$. Substituting $\lambda = 4$ and $k = 7$, we get $P(X=7) = \frac{e^{-4} 4^7}{7!}$, which calculates to approximately 0.0596.

Question 14

Compute Probabilities of a Poisson Random Variable
-A dictionary 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 three errors.
A) 0.0072
B) 0.0005
C) 0.1734
D) 0.0129

Accepted Answer

The answer of Compute Probabilities of a Poisson Random Variable
-A...

Question 15

Construct Binomial Probability Histograms
-Draw the probability histogram and label the mean for n = 8 and p = 0.3

Accepted Answer

The answer of Construct Binomial Probability Histograms
-Draw the probability histogram...

Question 16

Construct Binomial Probability Histograms
-Draw the probability histogram and label the mean for n = 7 and p = 0.5

Accepted Answer

The answer of Construct Binomial Probability Histograms
-Draw the probability histogram...

Question 17

Compute Probabilities of a Poisson Random Variable
-The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 8.2. Find the probability of observing exactly three accidents on this stretch of road next month.
A) 0.025239
B) 4.575166
C) 0.374135
D) 67.820284

Accepted Answer

The probability of observing exactly $k$ accidents when the events follow a Poisson distribution with mean $\lambda$ is given by $P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$. For $\lambda = 8.2$ and $k = 3$, $P(X=3) = \frac{e^{-8.2} * 8.2^3}{3!}$, which calculates to approximately 0.025239.

Question 18

Compute Probabilities of a Poisson Random Variable
-The number of road construction projects that take place at any one time in a certain city follows a Poisson distribution with a mean of 5. Find the probability that exactly three road construction projects are currently taking place in this city.
A) 0.140374
B) 0.051862
C) 0.272887
D) 0.100819

Accepted Answer

The probability of observing exactly $k$ events in a Poisson distribution is given by the formula $P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$, where $\lambda$ is the mean number of events. Here, $\lambda = 5$ and $k = 3$, so $P(X=3) = \frac{e^{-5} 5^3}{3!} = 0.140374$.

Question 19

Compute Probabilities of a Poisson Random Variable
-Suppose the number of babies born during an 8-hour shift at a hospital s maternity wing follows a Poisson distribution with a mean of 7 an hour. Find the probability that three babies are born during a particular 1-hour period in this maternity wing.
A) 0.052129
B) 0.046544
C) 0.006516
D) 0.000315

Accepted Answer

The probability of three babies being born during a particular 1-hour period in this maternity wing can be calculated using the Poisson distribution formula:P(X = k) = (e^-λ * λ^k) / k!where:λ is the mean number of babies born per hourk is the number of babies bornIn this case, λ = 7 and k = 3. Plugging these values into the formula, we get:P(X = 3) = (e^-7 * 7^3) / 3! = 0.052129Therefore, the probability that three babies are born during a particular 1-hour period in this maternity wing is 0.052129.

Question 20

Compute Probabilities of a Poisson Random Variable
-The number of road construction projects that take place at any one time in a certain city follows a Poisson distribution with a mean of 2. Find the probability that more than four road construction projects are currently taking place in the city.
A) 0.052653
B) 0.142877
C) 0.947347
D) 0.857123

Accepted Answer

The probability of more than four road construction projects can be found by subtracting the probability of four or fewer projects from 1. Using the Poisson distribution formula, this involves calculating the cumulative probability for 0 to 4 projects and subtracting from 1. The correct calculation leads to a probability of approximately 0.052653 for more than four projects.

Quiz 6: Discrete Probability Distributions

Compute Probabilities of a Poisson Random Variable -The local police department receives an average of two calls per hour. Use the Poisson distribution to determine the probability that in a randomly selected hour the number of calls is five.

Compute Probabilities of a Poisson Random Variable -An online retailer receives an average of five orders per 500 hits on its website. If it gets 100 hits on its website, find the probability of receiving at least two orders. Use the Poisson distribution.

Construct Binomial Probability Histograms -Given that a random variable x, the number of successes, follows a Poisson process, then the number of successes in any interval is independent of the number of successes in any other interval provided the intervals

Compute Probabilities of a Poisson Random Variable -A local animal rescue organization receives an average of 0.55 rescue calls per hour. Use the Poisson distribution to find the probability that during a randomly selected hour, the organization will receive fewer than two calls.

Compute Probabilities of a Poisson Random Variable -A customer service 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.

Construct Binomial Probability Histograms -Given that a random variable x, the number of successes, follows a Poisson process, then the probability of success for any two intervals of the same size

Compute Probabilities of a Poisson Random Variable -The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 8.9. Find the probability that less than three accidents will occur next month on this stretch of road.

Construct Binomial Probability Histograms -Draw the probability histogram and label the mean for n = 9 and p = 0.7

Construct Binomial Probability Histograms -Given that a random variable x, the number of successes, follows a Poisson process, the probability of 2 or more successes in any sufficiently small subinterval is

Compute Probabilities of a Poisson Random Variable -A help desk 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 seven calls. Use the Poisson distribution.

Compute Probabilities of a Poisson Random Variable -A dictionary 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 three errors.

Construct Binomial Probability Histograms -Draw the probability histogram and label the mean for n = 8 and p = 0.3

Construct Binomial Probability Histograms -Draw the probability histogram and label the mean for n = 7 and p = 0.5

Compute Probabilities of a Poisson Random Variable -The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 8.2. Find the probability of observing exactly three accidents on this stretch of road next month.

Compute Probabilities of a Poisson Random Variable -The number of road construction projects that take place at any one time in a certain city follows a Poisson distribution with a mean of 5. Find the probability that exactly three road construction projects are currently taking place in this city.

Compute Probabilities of a Poisson Random Variable -The number of road construction projects that take place at any one time in a certain city follows a Poisson distribution with a mean of 2. Find the probability that more than four road construction projects are currently taking place in the city.