Question 1

Which of the following is NOT a requirement for a probability experiment to be classified as a binomial experiment?

Accepted Answer

Each trial must have more than two possible outcomes.

Question 2

A pharmaceutical company tests a new drug on 10 patients. Each patient has a 70% chance of experiencing a positive effect. What is the probability that exactly 7 patients experience a positive effect? Use the binomial probability formula.

Accepted Answer

$$P(x=7) = \binom{10}{7} (0.7)^7$ (0.3)^3$

Question 3

Which of the following scenarios is NOT a binomial experiment?

Accepted Answer

Drawing 3 cards from a deck without replacement and counting the number of aces.

Question 4

Given a binomial experiment with $$n = 12$$ and $$p = 0.4$$, what is the expected value (mean) of the number of successes?

Accepted Answer

$$\mu = n \cdot p = 12 \cdot 0.4 = 4.8$$

Question 5

A survey finds that 30% of adults exercise regularly. If 5 adults are randomly selected, what is the probability that at least 3 of them exercise regularly?

Accepted Answer

$$P(x \geq 3) = P(x=3) + P(x=4) + P(x=5)$$

Question 6

If $$n = 20$$, $$p = 0.1$$, and $$q = 0.9$$, what is the variance of the binomial distribution?

Accepted Answer

$$\$$sigma^2$$ = n p q = 20 \cdot 0.1 \cdot 0.9 = 1.8$$

Question 7

A researcher selects 8 people at random and asks if they have traveled outside the country in the past year. If the probability that a person has traveled is 0.25, what is the probability that exactly 2 people have traveled?

Accepted Answer

$$P(x=2) = \binom{8}{2} (0.25)^2$ (0.75)^6$

Question 8

Which of the following best describes the standard deviation of a binomial distribution?

Accepted Answer

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

Question 9

A city reports that 56% of days in June are cloudy. For a 30-day June, what is the probability that there are exactly 20 cloudy days? (Let $$p = 0.56$$ and $$n = 30.)$$

Accepted Answer

$$P(x=20) = \binom{30}{20} (0.56$$)^{20}$$ (0.44$$)^{10}$$

Question 10

In a binomial experiment with $$n = 15$$ and $$p = 0.2$$, what is the probability of getting zero successes?

Accepted Answer

$$P(x=0) = (0.8$$)^{15}$$

Question 11

A survey found that 22% of adults believe the economy is the most important issue. If 4 adults are selected at random, what is the probability that at least two of them believe this?

Accepted Answer

$$P(x \geq 2) = P(x=2) + P(x=3) + P(x=4)$$

Question 12

Which of the following statements about the sum of probabilities in a binomial distribution is correct?

Accepted Answer

The sum of all probabilities for all possible values of $$x is 1.$$

Question 13

A doctor performs a surgery with an 85% success rate on 8 patients. What are the possible values for the random variable $$x$$, the number of successful surgeries?

Accepted Answer

$$x = 0, 1, 2, 3, 4, 5, 6, 7, 8$$

Question 14

If the probability of success in a binomial experiment is $$p = 0.75$$ and the number of trials is $$n = 100$$, which of the following would be considered an unusual event?

Accepted Answer

Getting exactly 66 successes, since $$P(x=66) \u003c 0.05$$

Question 15

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

Accepted Answer

$$P(x) = \binom{n}{x} p^x (1-p$$)^{n-x}$$

Question 16

A random variable $$x in a $$binomial experiment represents:

Accepted Answer

The number of successes in $$n$$ trials.

Question 17

If a binomial experiment has $$n = 6$$ and $$p = 0.64$$, what is the probability that exactly 4 people out of 6 are 65 years or older?

Accepted Answer

$$P(x=4) = \binom{6}{4} (0.64)^4$ (0.36)^2$

Question 18

Which of the following is a correct interpretation of the standard deviation in a binomial distribution?

Accepted Answer

It measures the typical deviation from the mean number of successes.

Binomial Distributions: Concepts, Calculations, and Applications

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