Q: Given a binomial distribution with $$n = 12$$ and $$p = 0.3$$, what is the mean $$M of $$the distribution?

$$M = n \times p = 12 \times 0.3 = 3.6$$

Q: A fair coin is flipped 5 times. What is the probability of getting exactly 3 heads?

$$P(3) = \frac{5!}{3!2!} (0.5)^3$ (0.5)^2$ = 10 \times 0.125 \times 0.25 = 0.3125$$

Q: A quality control inspector checks 20 light bulbs, each with a 2% chance of being defective. What is the expected number of defective bulbs?

$$M = n p = 20 \times 0.02 = 0.4$$

Question 1

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

Accepted Answer

Each trial has more than two possible outcomes.

Question 2

Given a binomial distribution with $$n = 12$$ and $$p = 0.3$$, what is the mean $$M of $$the distribution?

Accepted Answer

$$M = n 	imes p = 12 	imes 0.3 = 3.6$$

Question 3

Which formula correctly gives the probability of exactly $$x$$ successes in $$n$$ binomial trials?

Accepted Answer

$$P(x) = \frac{n!}{x!(n-x)!} p^x q$$^{n-x}$$

Question 4

A fair coin is flipped 5 times. What is the probability of getting exactly 3 heads?

Accepted Answer

$$P(3) = \frac{5!}{3!2!} (0.5)^3$ (0.5)^2$ = 10 	imes 0.125 	imes 0.25 = 0.3125$$

Question 5

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

Accepted Answer

Counting the number of emails received per hour.

Question 6

For a Poisson distribution, if the average number of occurrences per interval is $$M = 4$$, what is the standard deviation $$\sigma$$?

Accepted Answer

$$\sigma = \sqrt{M} = \sqrt{4} = 2$$

Question 7

Which formula gives the probability of observing exactly $$x$$ occurrences in a Poisson distribution with mean $$M$$?

Accepted Answer

$$P(x) = \frac{M^x e$$^{-M}$$}{x!}$$

Question 8

When can the binomial distribution be approximated by the Poisson distribution?

Accepted Answer

When $$n is $$large and $$p is $$small such that $$np  100.$$

Question 9

A hospital records an average of 3 births per hour. What is the probability that exactly 5 births occur in a given hour? (Use the Poisson formula.)

Accepted Answer

$$P(5) = \frac{3^5$ e$$^{-3}$$}{5!}$$

Question 10

Which of the following is the correct formula for the standard deviation of a binomial distribution?

Accepted Answer

$$\sigma = \sqrt{n p q}$$

Question 11

A quality control inspector checks 20 light bulbs, each with a 2% chance of being defective. What is the expected number of defective bulbs?

Accepted Answer

$$M = n p = 20 	imes 0.02 = 0.4$$

Question 12

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

Accepted Answer

The number of trials is fixed.

Question 13

If $$n = 7$$ and $$p = 0.4 in a $$binomial distribution, what is the probability of exactly 2 successes?

Accepted Answer

$$P(2) = \frac{7!}{2!5!} (0.4)^2$ (0.6)^5$

Question 14

Which of the following best describes the sample space in a probability experiment?

Accepted Answer

The set of all possible outcomes.

Question 15

A rare disease occurs with probability $$p = 0.001 in a $$population of $$n = 2000. $$Which distribution is most appropriate to model the number of cases?

Accepted Answer

Poisson distribution, since $$n is $$large and $$p is $$small.

Binomial and Poisson Distributions: Study Notes for Introductory Statistics

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