Q: A school raffle has 1500 tickets and 800 students each buy one ticket. The chance of winning a prize is $$p = \frac{1}{1500}. $$Which distribution should be used to approximate the probability that exactly 2 students win prizes?

Poisson distribution with $$\lambda = 800 \times \frac{1}{1500}$$

Q: A population of 15,000 mice has an independent probability of 0.002 of carrying a rare mutation. What is the expected number of mice carrying the mutation?

$$\lambda = 15,000 \times 0.002 = 30$$

Q: If the mean number of cars passing through an intersection in 10 minutes is 17.4, what is the probability that at most 15 cars pass through in 10 minutes? (Use cumulative Poisson probability notation.)

$$P(X \leq 15) = \text{poissoncdf}(17.4, 15)$$

Question 1

Which of the following scenarios is most appropriately modeled by a Poisson distribution?

Accepted Answer

Counting the number of cars passing through an intersection in a 10-minute interval, given the average rate is known.

Question 2

A bakery receives on average 2 customers every 15 minutes. What is the probability that exactly 3 customers enter in a 15-minute period? Use the Poisson formula.

Accepted Answer

$$P(X=3) = \frac{2^3$ e$$^{-2}$$}{3!}$$

Question 3

Which condition is NOT required for a process to be modeled by a Poisson distribution?

Accepted Answer

The number of trials is fixed.

Question 4

A textile inspector finds on average 0.5 defects per meter of fabric. What is the probability that she finds 0 defects in a 2-meter length?

Accepted Answer

$$P(X=0) = \frac{(1)^0$ e$$^{-1}$$}{0!}$$

Question 5

Which formula gives the probability of observing $$k$$ events in a Poisson process with mean rate $$\lambda$$?

Accepted Answer

$$P(X=k) = \frac{\lambda^k e$$^{-\lambda}$$}{k!}$$

Question 6

A school raffle has 1500 tickets and 800 students each buy one ticket. The chance of winning a prize is $$p = \frac{1}{1500}. $$Which distribution should be used to approximate the probability that exactly 2 students win prizes?

Accepted Answer

Poisson distribution with $$\lambda = 800 	imes \frac{1}{1500}$$

Question 7

A customer service call center receives an average of 12 calls per hour. What is the probability that exactly 20 calls are received in a 2-hour period?

Accepted Answer

$$P(X=20) = \frac{24$$^{20}$$ e$$^{-24}$$}{20!}$$

Question 8

Which of the following is a key difference between the Binomial and Poisson distributions?

Accepted Answer

The Binomial distribution models the number of successes in a fixed number of trials, while the Poisson models the number of events in a fixed interval.

Question 9

A population of 15,000 mice has an independent probability of 0.002 of carrying a rare mutation. What is the expected number of mice carrying the mutation?

Accepted Answer

$$\lambda = 15,000 	imes 0.002 = 30$$

Question 10

If the mean number of cars passing through an intersection in 10 minutes is 17.4, what is the probability that at most 15 cars pass through in 10 minutes? (Use cumulative Poisson probability notation.)

Accepted Answer

$$P(X \leq 15) = 	ext{poissoncdf}(17.4, 15)$$

Question 11

A student observes an average of 3.6 birds landing on a feeder per hour. What is the standard deviation of the number of birds landing in one hour?

Accepted Answer

$$\sqrt{3.6}$$

Question 12

Which of the following is TRUE about the Poisson distribution?

Accepted Answer

The mean and variance of the Poisson distribution are equal.

Question 13

A call center receives an average of 12 calls per hour. What is the probability that fewer than 8 calls are received in a 30-minute period? (Use cumulative Poisson probability notation.)

Accepted Answer

$$P(X \u003c 8) = 	ext{poissoncdf}(6, 7)$$

Question 14

A Poisson process has a mean rate $$\lambda = 4$$ events per hour. What is the probability of observing 2 or fewer events in one hour?

Accepted Answer

$$P(X \leq 2) = 	ext{poissoncdf}(4, 2)$$

Question 15

Which of the following is a valid use of the Poisson distribution to approximate the Binomial distribution?

Accepted Answer

When $$n is $$large and $$p is $$small, with $$np = \lambda.$$

Question 16

A student working on a transportation engineering project analyzes traffic at an intersection. The average number of cars per minute is 17.4. What is the probability that exactly 15 cars go through the intersection during the first minute?

Accepted Answer

$$P(X=15) = 	ext{poissonpdf}(17.4, 15)$$

Question 17

A Poisson process has a mean rate of 0.5 defects per meter. What is the probability that an inspector finds exactly 3 defects in 4 meters?

Accepted Answer

$$P(X=3) = \frac{2^3$ e$$^{-2}$$}{3!}$$

Question 18

A rare mutation occurs in mice with probability 0.002. In a population of 15,000 mice, what is the probability that at most 2 mice carry the mutation? (Use cumulative Poisson probability notation.)

Accepted Answer

$$P(X \leq 2) = 	ext{poissoncdf}(30, 2)$$

Poisson Distribution: Concepts, Applications, and Calculations

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