Question 1

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

Accepted Answer

The number of trials must be fixed.

Question 2

Suppose a survey finds that 8% of adults are left-handed. If 12 adults are randomly selected, what is the probability that exactly 2 are left-handed? Use the binomial probability formula.

Accepted Answer

$$P(2) = \dfrac{12!}{2!\,10!} \cdot 0.08^2$ \cdot 0.92$$^{10}$$ = 0.233$$

Question 3

Given a binomial distribution with $$n = 20$$ and $$p = 0.3$$, what is the mean and standard deviation?

Accepted Answer

Mean: $$\mu = 6$$, Standard deviation: $$\sigma = 2.05$$

Question 4

A quality control inspector checks 15 light bulbs, each with a 0.1 probability of being defective. What is the probability that none are defective?

Accepted Answer

$$P(0) = \dfrac{15!}{0!\,15!} \cdot 0.1^0$ \cdot 0.9$$^{15}$$ = 0.205$$

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

If the mean number of car accidents at an intersection per month is 2.5, what is the probability that exactly 4 accidents occur in a given month? Use the Poisson formula.

Accepted Answer

$$P(4) = \dfrac{2.5^4$ \cdot e$$^{-2.5}$$}{4!} = 0.133$$

Question 7

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

Accepted Answer

The mean and variance are equal.

Question 8

A hospital records an average of 3 births per hour. What is the standard deviation for the number of births per hour?

Accepted Answer

$$\sigma = \sqrt{3} = 1.732$$

Question 9

In a binomial experiment with $$n = 100$$ and $$p = 0.02$$, which distribution can be used as an approximation and why?

Accepted Answer

Poisson, because $$n is $$large and $$p is $$small.

Question 10

A rare disease occurs with a probability of 0.001 in the population. If 2000 people are tested, what is the expected number of cases and which distribution is most appropriate to model the number of cases?

Accepted Answer

Expected cases: $$2$$, Poisson distribution.

Question 11

A study finds that the probability a randomly selected person is a smoker is 0.18. In a group of 8 people, what is the probability that at least one is a smoker?

Accepted Answer

$$1 - (0.82^8$) = 0.771$$

Question 12

If the mean number of hurricanes per year is 5.5, what is the probability that in a randomly selected year there are exactly 6 hurricanes? (Use $$e = 2.71828)$$

Accepted Answer

$$P(6) = \dfrac{5.5^6$ \cdot 2.71828$$^{-5.5}$$}{6!} = 0.157$$

Question 13

Which of the following values would be considered significantly high for a binomial distribution with $$n = 460$$, $$p = 0.5$$? (Use the range rule of thumb.)

Accepted Answer

252

Question 14

A binomial experiment has $$n = 15$$ and $$p = 0.4. $$What is the variance?

Accepted Answer

$$\$$sigma^2$$ = npq = 15 	imes 0.4 	imes 0.6 = 3.6$$

Question 15

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

Accepted Answer

The number of occurrences must be fixed.

Question 16

A lottery game has a probability of winning of $$p = 0.0001$$ per ticket. If you buy 500 tickets, what is the approximate probability of winning at least once? (Use the Poisson approximation.)

Accepted Answer

$$1 - e$$^{-0.05}$$ = 0.0488$$

Question 17

Which formula gives the probability of exactly $$x$$ successes in $$n$$ binomial trials with probability $$p of $$success?

Accepted Answer

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

Chapter 5: Probability Distributions – Binomial and Poisson Distributions

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