Question 1

Given that a random variable x, the number of successes, follows a Poisson process, then the number ofsuccesses 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 defining property of a Poisson process is that the number of events in disjoint (non-overlapping) intervals are independent of each other. This means that what happens in one interval does not affect what happens in another, provided the intervals do not overlap.

Question 2

A dictionary contains 500 pages. If there are 200 typing errors randomly distributed throughout the book, usethe 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 Poisson distribution is given by:P(X = k) = (e^-λ * λ^k) / k!where:λ is the average number of events per unit time or spacek is the number of events that occurIn this case, λ = 200 / 500 = 0.4.So, the probability of a page containing exactly three errors is:P(X = 3) = (e^-0.4 * 0.4^3) / 3! = 0.0072Therefore, the correct answer is A.

Question 3

The probability that a house in an urban area will develop a leak is 5%. If 20 houses are randomly selected,what is the mean of the number of houses that developed leaks?
A) 1
B) 2
C) 0.5
D) 1.5

Accepted Answer

The mean of a binomial distribution is calculated as n * p, where n is the number of trials (houses) and p is the probability of success (developing a leak). Here, n = 20 and p = 0.05, so the mean is 20 * 0.05 = 1.

Question 4

A history professor finds that when he schedules an office hour at the 10:30 a.m. time slot, an average of threestudents arrive. Use the Poisson distribution to find the probability that in a randomly selected office hour inthe 10:30 a.m. time slot exactly seven students will arrive.
A) 0.0216
B) 0.0004
C) 0.0034
D) 0.0001

Accepted Answer

The Poisson distribution is given by the formula P(X = k) = (e^-λ * λ^k) / k!, where λ is the average number of arrivals. In this case, λ = 3, so the probability of exactly 7 students arriving is P(X = 7) = (e^-3 * 3^7) / 7! = 0.0216.

Question 5

The number of traffic accidents that occur on a particular stretch of road during a month follows a Poissondistribution with a mean of 7.5. Find the probability of observing exactly five accidents on this stretch of roadnext month.
A) 0.109375
B) 1.332455
C) 0.805166
D) 9.808925

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 $\lambda = 7.5$ and $k = 5$, the probability is $P(X=5) = \frac{e^{-7.5} * 7.5^5}{5!}$, which calculates to approximately 0.109375.

Question 6

In a recent survey, 80% of the community favored building a health center in their neighborhood. If 15 citizensare chosen, what is the standard deviation of the number favoring the health center?
A) 1.55
B) 2.40
C) 0.98
D) 0.55

Accepted Answer

The standard deviation for a binomial distribution is calculated using the formula √(n * p * (1-p)), where n is the number of trials, and p is the probability of success. Here, n = 15 and p = 0.8, so the standard deviation is √(15 * 0.8 * 0.2) = 1.55.

Question 7

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 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 probability of receiving less than three calls is given by the sum of the probabilities of receiving 0, 1, or 2 calls. The probability of receiving 0 calls is e^-3, the probability of receiving 1 call is 3e^-3/1!, and the probability of receiving 2 calls is 9e^-3/2!. Therefore, the probability of receiving less than three calls is e^-3 + 3e^-3 + 9e^-3/2! = 0.4232. The probability of receiving at least three calls is therefore 1 - 0.4232 = 0.5768.

Question 8

The number of road construction projects that take place at any one time in a certain city follows a Poissondistribution with a mean of 5. Find the probability that exactly six road construction projects are currentlytaking place in this city.
A) 0.146223
B) 0.322754
C) 0.072770
D) 0.160623

Accepted Answer

The probability of exactly k events in a Poisson distribution is given by $P(k) = \frac{\lambda^k e^{-\lambda}}{k!}$, where $\lambda$ is the mean number of events. Here, $\lambda = 5$ and $k = 6$, so $P(6) = \frac{5^6 e^{-5}}{6!} \approx 0.146223$.

Question 9

Given that a random variable x, the number of successes, follows a Poisson process, the probability of 2 or moresuccesses 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 event occurring in a very small interval is approximately zero.

Question 10

Draw the probability histogram and label the mean for n = 9 and p = 0.76.3 The Poisson Probability Distribution1 Determine if a probability experiment follows a Poisson process.

Accepted Answer

The answer of Draw the probability histogram and label the...

Question 11

The number of road construction projects that take place at any one time in a certain city follows a Poissondistribution with a mean of 6. Find the probability that more than four road construction projects are currentlytaking place in the city.
A) 0.714943
B) 0.848796
C) 0.285057
D) 0.151204

Accepted Answer

The probability of more than four projects is 1 minus the probability of four or fewer projects. Using the Poisson distribution with mean 6, P(X > 4) = 1 - P(X ≤ 4) = 1 - (e^(-6) * (6^0/0! + 6^1/1! + 6^2/2! + 6^3/3! + 6^4/4!)) = 0.714943.

Question 12

Given that a random variable x, the number of successes, follows a Poisson process, then the probability ofsuccess 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 a given number of events happening in a fixed interval of time or space is constant across identical intervals, assuming the process is homogeneous (the rate of occurrence is constant throughout the process).

Question 13

Draw the probability histogram and label the mean for n = 7 and p = 0.5

Accepted Answer

The answer of Draw the probability histogram and label the...

Question 14

Draw the probability histogram and label the mean for n = 8 and p = 0.3

Accepted Answer

The answer of Draw the probability histogram and label the...

Question 15

Draw the probability histogram and label the mean for n = 6 and p = 0.4

Accepted Answer

The answer of Draw the probability histogram and label the...

Question 16

The number of traffic accidents that occur on a particular stretch of road during a month follows a Poissondistribution with a mean of 7.2. Find the probability that less than three accidents will occur next month on thisstretch of road.
A) 0.025474
B) 0.071917
C) 0.974526
D) 0.928083

Accepted Answer

The probability of less than three accidents occurring next month on this stretch of road is given by the Poisson distribution with a mean of 7.2:P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)= e^(-7.2) * 7.2^0 / 0! + e^(-7.2) * 7.2^1 / 1! + e^(-7.2) * 7.2^2 / 2!= 0.025474

Question 17

The local police department receives an average of two calls per hour. Use the Poisson distribution todetermine 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 getting exactly k events in a time period can be found using the Poisson distribution formula: P(k; λ) = (e^(-λ) * λ^k) / k!, where λ is the average number of events (2 calls per hour in this case) and k is the number of events of interest (5 calls). Plugging the values into the formula gives P(5; 2) = (e^(-2) * 2^5) / 5! = 0.0361.

Question 18

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 average rate of orders per 100 hits is 1 (since 5 orders per 500 hits translates to 1 order per 100 hits). Using the Poisson distribution with λ = 1, the probability of receiving at least two orders is 1 minus the probability of receiving 0 or 1 order. This calculation gives approximately 0.2642.

Question 19

A local animal rescue organization receives an average of 0.55 rescue calls per hour. Use the Poissondistribution to find the probability that during a randomly selected hour, the organization will receive fewerthan two calls.
A) 0.894
B) 0.106
C) 0.317
D) 0.087

Accepted Answer

The Poisson distribution formula for the probability of observing k events in an interval is given by $P(k; \lambda) = \frac{e^{-\lambda} \lambda^k}{k!}$, where $\lambda$ is the average number of events per interval, and $k$ is the number of occurrences. For fewer than two calls, we calculate the probability of receiving 0 or 1 calls. With $\lambda = 0.55$, $P(0) = \frac{e^{-0.55} \cdot 0.55^0}{0!} = e^{-0.55}$, and $P(1) = \frac{e^{-0.55} \cdot 0.55^1}{1!} = 0.55e^{-0.55}$. Adding these probabilities gives the total probability of receiving fewer than two calls, which is approximately 0.894.

Question 20

A help desk receives an average of four calls per hour on its toll-free number. For any given hour, find theprobability that it will receive exactly six calls. Use the Poisson distribution.
A) 0.1042
B) 0.0141
C) 725.7202
D) 0.0330

Accepted Answer

The probability of receiving exactly six calls in an hour, given an average rate ($\lambda$) of 4 calls per hour, can be calculated using the Poisson distribution formula: $P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}$. Substituting $\lambda = 4$ and $k = 6$, we get $P(X=6) = \frac{4^6 e^{-4}}{6!}$, which calculates to approximately 0.1042.

Quiz 6: Discrete Probability Distributions

Given that a random variable x, the number of successes, follows a Poisson process, then the number ofsuccesses in any interval is independent of the number of successes in any other interval provided the intervals

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

The probability that a house in an urban area will develop a leak is 5%. If 20 houses are randomly selected,what is the mean of the number of houses that developed leaks?

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

The number of traffic accidents that occur on a particular stretch of road during a month follows a Poissondistribution with a mean of 7.5. Find the probability of observing exactly five accidents on this stretch of roadnext month.

In a recent survey, 80% of the community favored building a health center in their neighborhood. If 15 citizensare chosen, what is the standard deviation of the number favoring the health center?

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.

The number of road construction projects that take place at any one time in a certain city follows a Poissondistribution with a mean of 5. Find the probability that exactly six road construction projects are currentlytaking place in this city.

Given that a random variable x, the number of successes, follows a Poisson process, the probability of 2 or moresuccesses in any sufficiently small subinterval is

Draw the probability histogram and label the mean for n = 9 and p = 0.76.3 The Poisson Probability Distribution1 Determine if a probability experiment follows a Poisson process.

The number of road construction projects that take place at any one time in a certain city follows a Poissondistribution with a mean of 6. Find the probability that more than four road construction projects are currentlytaking place in the city.

Given that a random variable x, the number of successes, follows a Poisson process, then the probability ofsuccess for any two intervals of the same size

Draw the probability histogram and label the mean for n = 7 and p = 0.5

Draw the probability histogram and label the mean for n = 8 and p = 0.3

Draw the probability histogram and label the mean for n = 6 and p = 0.4

The number of traffic accidents that occur on a particular stretch of road during a month follows a Poissondistribution with a mean of 7.2. Find the probability that less than three accidents will occur next month on thisstretch of road.

The local police department receives an average of two calls per hour. Use the Poisson distribution todetermine the probability that in a randomly selected hour the number of calls is five.

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 local animal rescue organization receives an average of 0.55 rescue calls per hour. Use the Poissondistribution to find the probability that during a randomly selected hour, the organization will receive fewerthan two calls.

A help desk receives an average of four calls per hour on its toll-free number. For any given hour, find theprobability that it will receive exactly six calls. Use the Poisson distribution.