Q: A company finds that the probability a new product succeeds is 0.3. What is the probability that the first success occurs on the third trial?

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

Q: A study finds that the probability a randomly selected business survives five years is 0.52. If four businesses are selected at random, what is the probability that the fourth business is the first to have succeeded?

$$P(X=4) = (0.48)^3$ \times 0.52$$

Question 1

Which of the following is a condition for a random variable to follow a geometric distribution?

Accepted Answer

The random variable represents the number of the trial in which the first success occurs.

Question 2

A company finds that the probability a new product succeeds is 0.3. What is the probability that the first success occurs on the third trial?

Accepted Answer

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

Question 3

Which of the following best describes a Poisson distribution?

Accepted Answer

It models the number of times an event occurs in a fixed interval of time, area, or volume.

Question 4

Suppose the mean number of emails a professor receives per hour is 5. What is the probability that exactly 7 emails are received in an hour? Use the Poisson formula.

Accepted Answer

$$P(X=7) = \frac{5^7$ e$$^{-5}$$}{7!}$$

Question 5

A hospital records an average of 2 births per hour. What is the probability that in a randomly selected hour, no births occur?

Accepted Answer

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

Question 6

Which of the following is NOT a requirement for a Poisson distribution?

Accepted Answer

The trials are repeated until a success occurs.

Question 7

A study finds that the probability a randomly selected business survives five years is 0.52. If four businesses are selected at random, what is the probability that the fourth business is the first to have succeeded?

Accepted Answer

$$P(X=4) = (0.48)^3$ 	imes 0.52$$

Question 8

If the average number of cars passing through a toll booth per minute is 8, what is the probability that exactly 10 cars pass through in a given minute?

Accepted Answer

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

Question 9

Which scenario is best modeled by a geometric distribution?

Accepted Answer

Counting the number of trials until the first defective item is found in a quality control process.

Question 10

A wildlife biologist finds that the average number of rabbits per acre is 3.6. What is the probability that exactly 7 rabbits are found on a randomly selected acre?

Accepted Answer

$$P(X=7) = \frac{3.6^7$ e$$^{-3.6}$$}{7!}$$

Question 11

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 models the number of events in a fixed interval.

Question 12

A call center receives an average of 12 calls per hour. What is the probability that exactly 15 calls are received in an hour?

Accepted Answer

$$P(X=15) = \frac{12$$^{15}$$ e$$^{-12}$$}{15!}$$

Question 13

Which of the following is a property of the geometric distribution?

Accepted Answer

The probability of success is constant for each trial.

Question 14

A researcher is interested in the number of phone calls received at a help desk in a 10-minute interval. Which distribution should be used to model this scenario?

Accepted Answer

Poisson distribution

Question 15

If the probability of success in a geometric distribution is $$p$$, what is the probability that the first success occurs on the $$xth $$trial?

Accepted Answer

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

Discrete Probability Distributions: Geometric and Poisson Distributions

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