Q: If $$n = 12$$ and $$p = 0.4$$, what is the mean (expected value) of the binomial random variable $$X$$?

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

Q: Given a binomial experiment with $$n = 20$$ and $$p = 0.7$$, what is the standard deviation of the number of successes?

$$\sigma = \sqrt{n p (1-p)} = \sqrt{20 \times 0.7 \times 0.3}$$

Question 1

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

Accepted Answer

Each trial has more than two possible outcomes.

Question 2

In a binomial probability distribution, what does the parameter $$p$$ represent?

Accepted Answer

The probability of success on a single trial.

Question 3

Suppose you flip a fair coin 8 times. What is the probability of getting exactly 3 heads? (Use the binomial probability formula.)

Accepted Answer

$$P(X=3) = \binom{8}{3} (0.5)^3$ (0.5$$)^{5}$$

Question 4

If $$n = 12$$ and $$p = 0.4$$, what is the mean (expected value) of the binomial random variable $$X$$?

Accepted Answer

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

Question 5

Which of the following scenarios is NOT a binomial experiment?

Accepted Answer

Drawing 5 cards from a deck without replacement and counting the number of aces.

Question 6

Given a binomial experiment with $$n = 20$$ and $$p = 0.7$$, what is the standard deviation of the number of successes?

Accepted Answer

$$\sigma = \sqrt{n p (1-p)} = \sqrt{20 	imes 0.7 	imes 0.3}$$

Question 7

A recent survey found that 65% of smartphone owners use a certain app. If 15 owners are randomly selected, what is the probability that exactly 10 use the app?

Accepted Answer

$$P(X=10) = \binom{15}{10} (0.65$$)^{10}$$ (0.35)^5$

Question 8

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

Accepted Answer

Approximately symmetric and bell-shaped.

Question 9

If $$p \u003c 0.5 in a $$binomial distribution, what is the shape of the distribution?

Accepted Answer

Skewed right.

Question 10

A binomial random variable $$X$$ counts the number of successes in $$n$$ independent trials. What are the possible values of $$X$$?

Accepted Answer

$$0, 1, 2, \ldots, n$$

Question 11

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

Accepted Answer

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

Question 12

In a binomial experiment, which of the following best describes the trials?

Accepted Answer

Each trial is independent of the others.

Question 13

A medical test has a 90% chance of correctly identifying a disease. If 8 patients with the disease are tested, what is the probability that exactly 7 are correctly identified?

Accepted Answer

$$P(X=7) = \binom{8}{7} (0.9)^7$ (0.1)^1$

Question 14

If $$n = 50$$ and $$p = 0.2$$, what is the standard deviation of the binomial random variable $$X$$?

Accepted Answer

$$\sigma = \sqrt{50 	imes 0.2 	imes 0.8}$$

Question 15

Which of the following is a real-world example of a binomial experiment?

Accepted Answer

Counting the number of defective products in a batch of 100, where each product is tested independently.

Question 16

If a binomial probability distribution is approximately bell-shaped, which of the following is most likely true?

Accepted Answer

$$p is $$close to 0.5 and $$n is $$large.

Question 17

A binomial experiment consists of 10 trials with $$p = 0.8. $$What is the probability of getting at least 8 successes?

Accepted Answer

$$P(X \geq 8) = P(X=8) + P(X=9) + P(X=10)$$

Question 18

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

Accepted Answer

The probability of success can change from trial to trial.

Question 19

If $$n = 25$$ and $$p = 0.6$$, what is the expected number of successes?

Accepted Answer

$$\mu = 25 	imes 0.6 = 15$$

Question 20

A binomial probability distribution is skewed left when:

Accepted Answer

$$p \u003e 0.5$$

Section 6.2: The Binomial Probability Distribution – Study Notes

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