Q: For a binomial experiment with $$n = 10$$ and $$p = 0.72$$, what is the expected number of adults who would rather give up chocolate?

$$E[X] = n p = 10 \times 0.72 = 7.2$$

Q: What is the variance of the number of adults who would rather give up chocolate in a sample of 10, given $$p = 0.72$$?

$$Var(X) = n p (1-p) = 10 \times 0.72 \times 0.28 = 2.016$$

Question 1

Which of the following scenarios can be modeled using a binomial probability distribution?

Accepted Answer

Counting the number of defective items in a batch of 20, where each item is independently defective with probability $$p.$$

Question 2

A probability experiment involves drawing three cards from a deck without replacement and recording the number of aces. Why does this scenario NOT fit the requirements for a binomial probability distribution?

Accepted Answer

The probability of drawing an ace changes after each draw, violating the independence assumption.

Question 3

Which formula is used to calculate the probability of exactly $$k$$ successes in $$n$$ independent binomial trials, each with probability $$p of $$success?

Accepted Answer

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

Question 4

According to a survey, 72% of adult Americans would rather give up chocolate than their cell phone. In a random sample of 10 adults, what is the probability that exactly 8 would rather give up chocolate?

Accepted Answer

$$P(X = 8) = \binom{10}{8} (0.72)^8$ (0.28)^2$

Question 5

In the same survey, what is the probability that at least 3 out of 10 adults would rather give up chocolate?

Accepted Answer

$$P(X \geq 3) = 1 - \sum_{k=0}^{2} \binom{10}{k} (0.72)^k (0.28$$)^{10-k}$$

Question 6

For a binomial experiment with $$n = 10$$ and $$p = 0.72$$, what is the expected number of adults who would rather give up chocolate?

Accepted Answer

$$E[X] = n p = 10 	imes 0.72 = 7.2$$

Question 7

What is the variance of the number of adults who would rather give up chocolate in a sample of 10, given $$p = 0.72$$?

Accepted Answer

$$Var(X) = n p (1-p) = 10 	imes 0.72 	imes 0.28 = 2.016$$

Question 8

If $$X is a $$binomial random variable with parameters $$n = 10$$ and $$p = 0.72$$, what is the probability that the number of adults who would rather give up chocolate is between 5 and 7 inclusive?

Accepted Answer

$$P(5 \leq X \leq 7) = \sum_{k=5}^{7} \binom{10}{k} (0.72)^k (0.28$$)^{10-k}$$

Question 9

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

Accepted Answer

Each trial must have more than two possible outcomes.

Question 10

A researcher wants to use technology to calculate binomial probabilities. Which of the following is a valid approach?

Accepted Answer

Using statistical software such as StatCrunch to compute $$P(X = k)$$ for given $$n$$, $$p$$, and $$k.$$

Question 11

Suppose you want to find the probability that at least 7 out of 10 adults would rather give up chocolate. Which expression correctly represents this probability?

Accepted Answer

$$P(X \geq 7) = \sum_{k=7}^{10} \binom{10}{k} (0.72)^k (0.28$$)^{10-k}$$

Question 12

Which of the following statements about the binomial probability distribution is TRUE?

Accepted Answer

The mean of a binomial distribution is $$n p$$ and the variance is $$n p (1-p).$$

Question 13

A binomial experiment is conducted with $$n = 10$$ and $$p = 0.72. $$What is the probability that none of the adults would rather give up chocolate?

Accepted Answer

$$P(X = 0) = (0.28$$)^{10}$$

Question 14

If the probability of success in a binomial experiment is $$p = 0.72$$, what is the probability of failure?

Accepted Answer

$$1 - p = 0.28$$

Question 15

Which of the following best describes the shape of a binomial probability distribution when $$p = 0.5$$ and $$n is $$large?

Accepted Answer

It is approximately symmetric and bell-shaped.

Question 16

A binomial probability table can be used to:

Accepted Answer

Quickly find probabilities for specific values of $$n$$, $$p$$, and $$k$$ without manual calculation.

The Binomial Probability Distribution: Concepts, Formulas, and Applications

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