Q: A pharmaceutical company tests a vaccine on 120 people. The probability of immunity is 80%. What is the expected number of people who will be immune?

$$E(X) = n \cdot p = 120 \times 0.8 = 96$$

Q: If $$X is a $$binomial random variable with $$n = 10$$ and $$p = 0.4$$, what is the standard deviation of $$X$$?

$$\sigma = \sqrt{n p (1-p)} = \sqrt{10 \times 0.4 \times 0.6} \approx 1.55$$

Q: A company finds that 60% of customers are expected to purchase a new product after seeing an advertisement. If 10 customers are surveyed, what is the variance of the number who purchase?

$$\text{Var}(X) = n p (1-p) = 10 \times 0.6 \times 0.4 = 2.4$$

Q: A company produces light bulbs with a defect rate of 5%. If 1000 bulbs are tested, what is the expected number of defective bulbs?

$$E(X) = 1000 \times 0.05 = 50$$

Question 1

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

Accepted Answer

Each trial must have more than two possible outcomes.

Question 2

A basketball player has a 70% chance of making a free throw. If she shoots 5 times, what is the probability that she makes exactly 3 shots? Use the binomial probability formula.

Accepted Answer

$$P(X = 3) = \binom{5}{3} (0.7)^3$ (0.3)^2$

Question 3

Which of the following scenarios is NOT a binomial experiment?

Accepted Answer

Drawing a card from a deck, recording its suit, and repeating without replacement.

Question 4

A pharmaceutical company tests a vaccine on 120 people. The probability of immunity is 80%. What is the expected number of people who will be immune?

Accepted Answer

$$E(X) = n \cdot p = 120 	imes 0.8 = 96$$

Question 5

If $$X is a $$binomial random variable with $$n = 10$$ and $$p = 0.4$$, what is the standard deviation of $$X$$?

Accepted Answer

$$\sigma = \sqrt{n p (1-p)} = \sqrt{10 	imes 0.4 	imes 0.6} \approx 1.55$$

Question 6

A gardener plants 8 seeds, each with a 65% chance of germinating. What is the probability that fewer than 4 seeds germinate?

Accepted Answer

$$P(X \u003c 4) = \sum_{x=0}^{3} \binom{8}{x} (0.65)^x (0.35$$)^{8-x}$$

Question 7

A company finds that 60% of customers are expected to purchase a new product after seeing an advertisement. If 10 customers are surveyed, what is the variance of the number who purchase?

Accepted Answer

$$	ext{Var}(X) = n p (1-p) = 10 	imes 0.6 	imes 0.4 = 2.4$$

Question 8

A national survey indicates that 38% of people have been in a car accident in the last 5 years. In a random sample of 10 people, what is the probability that exactly 4 have had an accident?

Accepted Answer

$$P(X = 4) = \binom{10}{4} (0.38)^4$ (0.62)^6$

Question 9

You take a 6-question quiz with True/False questions. What is the probability of getting all 6 questions correct by guessing?

Accepted Answer

$$P(X = 6) = (0.5)^6$

Question 10

A marketing company sends a promotional email to 10 customers. If 42% typically open the email, what is the probability that at least 1 customer opens it?

Accepted Answer

$$P(X \geq 1) = 1 - (0.58$$)^{10}$$

Question 11

A company produces light bulbs with a defect rate of 5%. If 1000 bulbs are tested, what is the expected number of defective bulbs?

Accepted Answer

$$E(X) = 1000 	imes 0.05 = 50$$

Question 12

If $$X is a $$binomial random variable with $$n = 20$$ and $$p = 0.3$$, what is $$P(X = 0)$$?

Accepted Answer

$$P(X = 0) = (0.7$$)^{20}$$

Question 13

A researcher is testing a new vaccine that is known to be 90% effective. In a clinical trial of 8 patients, what is the probability that exactly 7 are protected?

Accepted Answer

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

Question 14

A company tracks the quality of its customer service calls. If 90% of calls are rated as satisfactory, what is the probability that 18 or fewer out of 20 randomly selected calls are satisfactory?

Accepted Answer

$$P(X \leq 18) = \sum_{x=0}^{18} \binom{20}{x} (0.9)^x (0.1$$)^{20-x}$$

Question 15

Which of the following best describes the mean and standard deviation of a binomial distribution with parameters $$n$$ and $$p$$?

Accepted Answer

Mean: $$\mu = n p$$, Standard deviation: $$\sigma = \sqrt{n p (1-p)}$$

Question 16

A student guesses on all 10 questions of a multiple-choice test, each with 4 options. What is the probability that the student gets exactly 2 questions correct?

Accepted Answer

$$P(X = 2) = \binom{10}{2} (0.25)^2$ (0.75)^8$

Question 17

If a binomial experiment has $$n = 12$$ and $$p = 0.5$$, what is the probability of getting at least 10 successes?

Accepted Answer

$$P(X \geq 10) = \sum_{x=10}^{12} \binom{12}{x} (0.5)^x (0.5$$)^{12-x}$$

Question 18

A manufacturing company finds that 5% of its light bulbs are defective. If 100 bulbs are tested, what is the probability that none are defective?

Accepted Answer

$$P(X = 0) = (0.95$$)^{100}$$

Question 19

Which of the following is the correct formula for the probability of getting exactly $$k$$ successes in $$n$$ independent Bernoulli trials with probability $$p of $$success?

Accepted Answer

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

Binomial Distribution: Concepts, Calculations, and Applications

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