Q: Suppose the probability that a new restaurant survives its first year is 0.7. What is the probability that the first restaurant to fail is the third one selected?

$$P(X=3) = (0.7$$)^{2}$$ \times 0.3$$

Q: If the probability of a machine failing on any given day is 0.1, what is the probability that it will fail for the first time on the fifth day?

$$P(X=5) = (0.9$$)^{4}$$ \times 0.1$$

Question 1

Which of the following is NOT a condition for a geometric distribution?

Accepted Answer

The probability of success changes from trial to trial.

Question 2

Suppose the probability that a new restaurant survives its first year is 0.7. What is the probability that the first restaurant to fail is the third one selected?

Accepted Answer

$$P(X=3) = (0.7$$)^{2}$$ 	imes 0.3$$

Question 3

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

Accepted Answer

Counting the number of emails received per hour.

Question 4

A call center receives an average of 5 calls per hour. What is the probability that exactly 2 calls are received in a given hour? (Use the Poisson formula.)

Accepted Answer

$$P(X=2) = \frac{5$$^{2}$$ e$$^{-5}$$}{2!}$$

Question 5

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

Accepted Answer

The geometric distribution models the number of trials until the first success, while the Poisson distribution models the number of events in a fixed interval.

Question 6

If the mean number of cars passing through a toll booth per minute is 4, what is the probability that exactly 6 cars pass through in a given minute?

Accepted Answer

$$P(X=6) = \frac{4$$^{6}$$ e$$^{-4}$$}{6!}$$

Question 7

A researcher is interested in the number of bacteria colonies that develop in a petri dish over a 24-hour period. Which probability distribution should she use?

Accepted Answer

Poisson distribution

Question 8

Which formula gives the probability that the first success occurs on the $$xth $$trial in a geometric distribution with probability of success $$p$$?

Accepted Answer

$$P(X=x) = (1-p$$)^{x-1}$$ p$$

Question 9

A hospital records an average of 2 births per hour. What is the probability that there are no births in a given hour?

Accepted Answer

$$P(X=0) = \frac{2$$^{0}$$ e$$^{-2}$$}{0!}$$

Question 10

Which of the following is NOT a property of the Poisson distribution?

Accepted Answer

The probability of an event occurring changes for each interval.

Question 11

A wildlife biologist finds that the average number of deer in a square mile is 2. What is the probability that exactly 5 deer are found in a randomly selected square mile?

Accepted Answer

$$P(X=5) = \frac{2$$^{5}$$ e$$^{-2}$$}{5!}$$

Question 12

If the probability of a machine failing on any given day is 0.1, what is the probability that it will fail for the first time on the fifth day?

Accepted Answer

$$P(X=5) = (0.9$$)^{4}$$ 	imes 0.1$$

Question 13

Which of the following best describes a real-life application of the geometric distribution?

Accepted Answer

Counting the number of coin tosses until the first heads appears.

Question 14

A city averages 10 power outages per year. What is the probability that there will be exactly 8 outages in a given year?

Accepted Answer

$$P(X=8) = \frac{10$$^{8}$$ e$$^{-10}$$}{8!}$$

Question 15

Which of the following is a valid use of the Poisson distribution?

Accepted Answer

Finding the probability of a certain number of phone calls received in a fixed time period.

Discrete Probability Distributions: Geometric and Poisson Distributions

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