Q: A survey finds that 30% of adults in a city own a pet cat. If 7 adults are randomly selected, what are the values of $$n$$, $$p$$, $$q$$, and $$x$$ for this binomial experiment?

$$n = 7, p = 0.3, q = 0.7, x = \text{number of adults who own a cat}$$

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

Suppose a factory produces light bulbs with a 5% defect rate. If you randomly select 10 bulbs, what is the probability that exactly 2 are defective? Use the binomial probability formula.

Accepted Answer

$$P(2) = (10C2)(0.05)^2$(0.95)^8 = 0.0746$

Question 3

A survey finds that 30% of adults in a city own a pet cat. If 7 adults are randomly selected, what are the values of $$n$$, $$p$$, $$q$$, and $$x$$ for this binomial experiment?

Accepted Answer

$$n = 7, p = 0.3, q = 0.7, x = 	ext{number of adults who own a cat}$$

Question 4

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

Accepted Answer

$$P(x) = (nCx) p^x q$$^{n-x}$$

Question 5

A recycling bin contains 6 items, and each item has a 20% chance of being non-recyclable. What is the probability that exactly 3 items are non-recyclable?

Accepted Answer

$$P(3) = (6C3)(0.2)^3$(0.8)^3 = 0.0512$

Question 6

In a binomial experiment with $$n = 8$$, $$p = 0.1$$, and $$q = 0.9$$, what is the probability that no more than 3 balloons are defective?

Accepted Answer

$$P(x \leq 3) = P(0) + P(1) + P(2) + P(3) = 0.43046721 + 0.38263752 + 0.14880348 + 0.03306744 = 0.99497565$$

Question 7

A telemarketer makes 10 calls, and each call has a 90% chance of resulting in a hang-up or no answer. What is the probability that more than 7 calls result in a hang-up or no answer?

Accepted Answer

$$P(x \u003e 7) = P(8) + P(9) + P(10) = 0.1937102445 + 0.3874204890 + 0.3486784401 = 0.9298091736$$

Question 8

Which of the following scenarios is NOT a binomial experiment?

Accepted Answer

Drawing cards from a deck without replacement and recording the number of red cards.

Question 9

Given a binomial distribution with $$n = 20$$, $$p = 0.41$$, and $$q = 0.59$$, what is the mean, variance, and standard deviation?

Accepted Answer

Mean: $$\mu = np = 8.2$$; Variance: $$\$$sigma^2$$ = npq = 4.8478$$; Standard deviation: $$\sigma = \sqrt{4.8478} = 2.2018$$

Question 10

A company finds that 15% of its products are returned. If 12 products are sold, what is the probability that at least 2 are returned?

Accepted Answer

$$P(x \geq 2) = 1 - [P(0) + P(1)] = 1 - [(12C0)(0.15)^0$(0.85)^{12}$$ + $$(12C1)(0.15)^1(0.85$$)^{11}$$]$$

Question 11

If $$p = 0.25$$ and $$n = 4$$, what is the probability distribution for $$x = 0, 1, 2, 3, 4$$ successes?

Accepted Answer

$$P(x) = (4Cx)(0.25)^x(0.75$$)^{4-x}$$ for x = 0, 1, 2, 3, 4$

Question 12

A student claims that the probability of getting exactly 2 heads in 4 flips of a fair coin is $$0.25. Is $$this correct? Why or why not?

Accepted Answer

No, the correct probability is $$P(2) = (4C2)(0.5)^2$(0.5)^2 = 0.375$

Question 13

Which calculator function would you use to find the probability of getting at most $$k$$ successes in a binomial experiment?

Accepted Answer

binomcdf

Question 14

If a probability distribution is graphed as a histogram and the probabilities are highest in the middle values of $$x$$, what does this suggest about the distribution?

Accepted Answer

The distribution is symmetric and most likely centered around the mean.

Question 15

A binomial experiment has $$n = 5$$ and $$p = 0.45. $$What is the probability of getting exactly 3 successes?

Accepted Answer

$$P(3) = (5C3)(0.45)^3$(0.55)^2 = 0.2757$

Binomial Probability Distributions: Concepts, Calculations, and Applications

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