Q: A company produces light bulbs with a defect rate of 5%. Out of 1000 bulbs, what is the expected number of defective bulbs and the standard deviation?

Expected: $$E(X) = 1000 \times 0.05 = 50$$; Std Dev: $$\sigma = \sqrt{1000 \times 0.05 \times 0.95} \approx 6.89$$

Question 1

Which of the following scenarios is a binomial experiment?

Accepted Answer

Flipping a coin 6 times and counting the number of heads.

Question 2

A basketball player has a 70% chance of making a free throw. If the player shoots 5 free throws, what is the probability that exactly 3 are successful? 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 is NOT a requirement for a binomial experiment?

Accepted Answer

The probability of success must change from trial to trial.

Question 4

A pharmaceutical company tests a vaccine on 120 patients. The probability of immunity is 80%. What is the expected number of patients who gain immunity?

Accepted Answer

$$E(X) = 120 	imes 0.8 = 96$$

Question 5

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) = \binom{6}{6} (0.5)^6$ (0.5)^0$

Question 6

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

Accepted Answer

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

Question 7

A company produces light bulbs with a defect rate of 5%. Out of 1000 bulbs, what is the expected number of defective bulbs and the standard deviation?

Accepted Answer

Expected: $$E(X) = 1000 	imes 0.05 = 50$$; Std Dev: $$\sigma = \sqrt{1000 	imes 0.05 	imes 0.95} \approx 6.89$$

Question 8

National surveys indicate that 38% of people have been in a car accident in the last 5 years. If you randomly sample 50 people, what is the probability that exactly 4 had an accident in the last 5 years?

Accepted Answer

$$P(X = 4) = \binom{50}{4} (0.38)^4$ (0.62$$)^{46}$$

Question 9

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 the email?

Accepted Answer

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

Question 10

Which formula gives the standard deviation of a binomial distribution with $$n$$ trials and probability $$p of $$success?

Accepted Answer

$$\sigma = \sqrt{n p (1-p)}$$

Question 11

A researcher tests a new vaccine on 10 patients. The vaccine is known to be 50% effective. What is the probability that the vaccine is effective for 7 or fewer patients?

Accepted Answer

$$P(X \leq 7) = \sum_{x=0}^{7} \binom{10}{x} (0.5)^x (0.5$$)^{10-x}$$

Question 12

A company tracks customer service calls. 90% of calls are rated as 'satisfied'. On a random day, 20 calls are made. What is the probability that 18 or fewer customers are satisfied?

Accepted Answer

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

Question 13

Which of the following best describes the meaning of $$P(X = k) in a $$binomial distribution?

Accepted Answer

The probability of getting exactly $$k$$ successes in $$n$$ independent trials.

Question 14

A sanctuary tests a bird species for improved nesting attempts. Each attempt has a 70% success rate. If 400 attempts are made, what is the expected number of successful attempts?

Accepted Answer

$$E(X) = 400 	imes 0.7 = 280$$

Question 15

If a binomial experiment has $$n = 12$$ trials and $$p = 0.4$$, what is the mean and standard deviation?

Accepted Answer

Mean: $$\mu = 12 	imes 0.4 = 4.8$$; Std Dev: $$\sigma = \sqrt{12 	imes 0.4 	imes 0.6} \approx 1.697$$

Question 16

A marketing company wants to know the probability that between 0 and 2 customers open a promotional email sent to 10 people, with a 42% open rate. Which formula should be used?

Accepted Answer

$$P(0 \leq X \leq 2) = \sum_{x=0}^{2} \binom{10}{x} (0.42)^x (0.58$$)^{10-x}$$

Question 17

Which of the following is a valid use of the complement rule in binomial probability?

Accepted Answer

To find $$P(X \geq 1)$$, calculate $$1 - P(X = 0).$$

Binomial Distribution: Concepts, Calculations, and Applications

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