Q: Given the following probability distribution for the number of bedrooms in homes for sale: $$P(1)=0.10$$, $$P(2)=0.30$$, $$P(3)=0.40$$, $$P(4)=0.10$$, $$P(5)=0.05$$, $$P(6)=0.05$$, what is the expected number of bedrooms?

$$E(X) = 1 \times 0.10 + 2 \times 0.30 + 3 \times 0.40 + 4 \times 0.10 + 5 \times 0.05 + 6 \times 0.05 = 2.85$$

Q: A bank surveys 3 customers. If 80% prefer Saturday opening, what is the probability that exactly 2 prefer Saturday? Use the binomial formula.

$$P(X=2) = \frac{3!}{2!1!} (0.8)^2$ (0.2)^1$ = 3 \times 0.64 \times 0.2 = 0.384$$

Q: If $$X is a $$binomial random variable with $$n=8$$ and $$p=0.8$$, what is the expected value $$E(X)$$?

$$E(X) = n \times p = 8 \times 0.8 = 6.4$$

Question 1

Which of the following is NOT a requirement for a random variable to follow a binomial distribution?

Accepted Answer

The number of trials is fixed.

Question 2

A company inspects 10 products and classifies each as defective or not. If the probability of a product being defective is 0.1, what is the probability that exactly 2 products are defective? Use the binomial formula.

Accepted Answer

$$P(X=2) = \frac{10!}{2!8!} (0.1)^2$ (0.9)^8$

Question 3

Given the following probability distribution for the number of bedrooms in homes for sale: $$P(1)=0.10$$, $$P(2)=0.30$$, $$P(3)=0.40$$, $$P(4)=0.10$$, $$P(5)=0.05$$, $$P(6)=0.05$$, what is the expected number of bedrooms?

Accepted Answer

$$E(X) = 1 	imes 0.10 + 2 	imes 0.30 + 3 	imes 0.40 + 4 	imes 0.10 + 5 	imes 0.05 + 6 	imes 0.05 = 2.85$$

Question 4

A game costs $$3 to play. You roll a fair six-sided die and win 1 $$times the value shown. What is your expected profit per game?

Accepted Answer

$$E(Profit) = (1+2+3+4+5+6)/6 - 3 = 0.5$$

Question 5

Which formula correctly calculates the standard deviation of a discrete random variable $$X$$ with possible values $$x_i$$ and probabilities $$P(x_i)$$?

Accepted Answer

$$\sigma = \sqrt{\sum [x_i - E(X)]^2 P(x_i)}$$

Question 6

A bank surveys 3 customers. If 80% prefer Saturday opening, what is the probability that exactly 2 prefer Saturday? Use the binomial formula.

Accepted Answer

$$P(X=2) = \frac{3!}{2!1!} (0.8)^2$ (0.2)^1$ = 3 	imes 0.64 	imes 0.2 = 0.384$$

Question 7

Which of the following scenarios best fits the use of a Poisson distribution?

Accepted Answer

Counting the number of emails received per hour.

Question 8

If $$X is a $$binomial random variable with $$n=8$$ and $$p=0.8$$, what is the expected value $$E(X)$$?

Accepted Answer

$$E(X) = n 	imes p = 8 	imes 0.8 = 6.4$$

Question 9

A shipment is accepted if at most 1 defect is found in a sample of 5, with a defect rate of 0.05. What is the probability of accepting a good shipment?

Accepted Answer

$$P(X \leq 1) = P(X=0) + P(X=1)$$ where $$X \sim Binomial(n=5, p=0.05)$$

Question 10

Which of the following is the correct formula for the probability of $$x$$ successes in $$n$$ binomial trials with probability $$p of $$success?

Accepted Answer

$$P(X=x) = \frac{n!}{x!(n-x)!} p^x (1-p$$)^{n-x}$$

Question 11

A marketing firm finds that 35% of people support a new regulation. If 10 people are polled, what is the probability that 3 or fewer support the regulation? (Use the cumulative binomial table.)

Accepted Answer

$$P(X \leq 3) = 0.5138$$

Question 12

Which of the following best describes the expected value in a business context?

Accepted Answer

The long-run average outcome if the experiment is repeated many times.

Question 13

Given the following investment options, which has the highest risk as measured by variance? (Refer to the provided tables.)

Accepted Answer

Investment 3

Question 14

If a random variable $$X$$ has the following probability distribution: $$P(10)=0.6$$, $$P(20)=0.3$$, $$P(30)=0.1$$, what is $$E(X)$$?

Accepted Answer

$$E(X) = 10 	imes 0.6 + 20 	imes 0.3 + 30 	imes 0.1 = 15$$

Question 15

A hypergeometric distribution is most appropriate when:

Accepted Answer

Sampling is without replacement from a finite population.

Question 16

Which of the following is a discrete random variable?

Accepted Answer

Number of complaints received per day.

Question 17

If the standard deviation of a discrete random variable is 1.2, what is the variance?

Accepted Answer

$$Variance = (1.2)^2$ = 1.44$$

Question 18

A company wants a high probability of accepting good shipments and a low probability of accepting bad shipments. Which statistical tool is most appropriate for designing their sampling plan?

Accepted Answer

Binomial distribution

Discrete Probability Distributions: Concepts, Calculations, and Applications

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