Q: A pharmaceutical company tests a new drug. The probability that a patient experiences a side effect is $$p = 0.1. $$What is the probability that the first patient to experience a side effect is the fourth patient tested?

$$0.1 \times 0.9$$^{3}$$ = 0.0729$$

Q: A call center receives an average of $$5$$ calls per hour. What is the probability that exactly $$3$$ calls are received in a given hour? Use the Poisson formula.

$$\frac{5$$^{3}$$ \times e$$^{-5}$$}{3!}$$

Q: In the geometric distribution, if the probability of success $$p = 0.25$$, what is the probability that the first success occurs on the second trial?

$$0.25 \times 0.75$$^{1}$$ = 0.1875$$

Q: A hospital records an average of $$2$$ births per hour. What is the probability that no births occur in a given hour?

$$\frac{2$$^{0}$$ \times e$$^{-2}$$}{0!} = e$$^{-2}$$

Q: A factory has a mean of $$4$$ defective products per day. What is the probability that exactly $$6$$ defective products are found in a day?

$$\frac{4$$^{6}$$ \times e$$^{-4}$$}{6!}$$

Q: A city records an average of $$7$$ power outages per year. What is the probability that there are fewer than $$3$$ outages in a year?

$$\sum_{x=0}^{2} \frac{7$$^{x}$$ \times e$$^{-7}$$}{x!}$$

Question 1

Which of the following best describes the random variable $$x in a $$geometric distribution?

Accepted Answer

The number of trials until the first success occurs.

Question 2

A pharmaceutical company tests a new drug. The probability that a patient experiences a side effect is $$p = 0.1. $$What is the probability that the first patient to experience a side effect is the fourth patient tested?

Accepted Answer

$$0.1 	imes 0.9$$^{3}$$ = 0.0729$$

Question 3

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

Accepted Answer

The experiment consists of counting the number of events in a fixed interval.

Question 4

A call center receives an average of $$5$$ calls per hour. What is the probability that exactly $$3$$ calls are received in a given hour? Use the Poisson formula.

Accepted Answer

$$\frac{5$$^{3}$$ 	imes e$$^{-5}$$}{3!}$$

Question 5

In the geometric distribution, if the probability of success $$p = 0.25$$, what is the probability that the first success occurs on the second trial?

Accepted Answer

$$0.25 	imes 0.75$$^{1}$$ = 0.1875$$

Question 6

Which scenario is best modeled by a Poisson distribution?

Accepted Answer

Counting the number of emails received in an hour.

Question 7

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

Accepted Answer

$$\frac{2$$^{0}$$ 	imes e$$^{-2}$$}{0!} = e$$^{-2}$$

Question 8

Which of the following statements about the Poisson distribution is FALSE?

Accepted Answer

The probability of success changes from interval to interval.

Question 9

A factory has a mean of $$4$$ defective products per day. What is the probability that exactly $$6$$ defective products are found in a day?

Accepted Answer

$$\frac{4$$^{6}$$ 	imes e$$^{-4}$$}{6!}$$

Question 10

In a geometric distribution, if the probability of success is $$p = 0.3$$, what is the expected number of trials until the first success?

Accepted Answer

$$\frac{1}{0.3} = 3.33$$

Question 11

A city records an average of $$7$$ power outages per year. What is the probability that there are fewer than $$3$$ outages in a year?

Accepted Answer

$$\sum_{x=0}^{2} \frac{7$$^{x}$$ 	imes e$$^{-7}$$}{x!}$$

Question 12

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

Accepted Answer

$$p 	imes q$$^{x-1}$$

Question 13

A researcher observes an average of $$\mu = 2$$ mutations per DNA strand. What is the probability that exactly $$1$$ mutation occurs in a strand?

Accepted Answer

$$\frac{2$$^{1}$$ 	imes e$$^{-2}$$}{1!}$$

Question 14

If the probability of a business failing after five years is $$0.2$$, what is the probability that the first failed business is the third one selected?

Accepted Answer

$$0.2 	imes 0.8$$^{2}$$ = 0.128$$

Question 15

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

Accepted Answer

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

Question 16

A traffic engineer observes an average of $$\mu = 3$$ accidents per month at an intersection. What is the probability that more than $$4$$ accidents occur in a month?

Accepted Answer

$$1 - \sum_{x=0}^{4} \frac{3$$^{x}$$ 	imes e$$^{-3}$$}{x!}$$

Question 17

Which of the following best describes the parameter $$\mu in $$the Poisson distribution?

Accepted Answer

The mean number of occurrences per interval.

Question 18

A rare disease occurs on average $$0.5$$ times per year in a population. What is the probability that the disease does not occur at all in a year?

Accepted Answer

$$\frac{0.5$$^{0}$$ 	imes e$$^{-0.5}$$}{0!} = e$$^{-0.5}$$

Question 19

If the probability of success in each trial is $$p = 0.4$$, what is the probability that the first success occurs on the fifth trial?

Accepted Answer

$$0.4 	imes 0.6$$^{4}$$ = 0.05184$$

Geometric and Poisson Distributions: Finding Probabilities

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