Q: A discrete random variable $$X$$ has the following probability distribution: $$P(X=0)=0.2$$, $$P(X=1)=0.5$$, $$P(X=2)=0.3. $$What is the expected value $$E(X)$$?

$$E(X) = 0 \times 0.2 + 1 \times 0.5 + 2 \times 0.3 = 1.1$$

Question 1

A company is launching a new product and wants to estimate the probability that at least one of two randomly selected customers will purchase it. If the probability that a customer purchases the product is 0.3, what is the probability that at least one of the two customers makes a purchase?

Accepted Answer

$$1 - (0.7 	imes 0.7)$$

Question 2

Which of the following statements about independent events is correct?

Accepted Answer

If $$P(A|B) = P(A)$$, then events A and B are independent.

Question 3

A discrete random variable $$X$$ has the following probability distribution: $$P(X=0)=0.2$$, $$P(X=1)=0.5$$, $$P(X=2)=0.3. $$What is the expected value $$E(X)$$?

Accepted Answer

$$E(X) = 0 	imes 0.2 + 1 	imes 0.5 + 2 	imes 0.3 = 1.1$$

Question 4

A call center receives an average of 4 calls per hour. What is the probability that exactly 2 calls are received in a given hour? (Use the Poisson distribution formula: $$P(X=k) = \frac{\lambda^k e$$^{-\lambda}$$}{k!}$$, where $$\lambda=4$$ and $$k=2.)$$

Accepted Answer

$$P(X=2) = \frac{4^2$ e$$^{-4}$$}{2!}$$

Question 5

A fair coin is flipped 5 times. What is the probability of getting exactly 3 heads? (Use the binomial formula: $$P(X=k) = {n \choose k} p^k (1-p$$)^{n-k}$$, where n=5$, k=3$, p=0.5$.)

Accepted Answer

$$P(X=3) = {5 \choose 3} (0.5)^3$ (0.5)^2$

Question 6

Which of the following is NOT a property of the normal distribution?

Accepted Answer

It is skewed to the right.

Question 7

A machine fills bottles with a mean of 500 ml and a standard deviation of 5 ml. What is the probability that a randomly selected bottle contains more than 510 ml? (Assume normality. Use $$Z = \frac{X-\mu}{\sigma}.)$$

Accepted Answer

$$P(X \u003e 510) = P\left(Z \u003e \frac{510-500}{5}\right)$$

Question 8

A sample of 36 students has a mean test score of 80 and a population standard deviation of 12. What is the standard error of the sample mean?

Accepted Answer

$$\frac{12}{\sqrt{36}} = 2$$

Question 9

Which of the following best describes a marginal probability?

Accepted Answer

The probability of a single event occurring, regardless of the outcome of other events.

Question 10

A survey finds that 60% of customers prefer brand A, and 40% prefer brand B. If two customers are selected at random, what is the probability that both prefer brand A?

Accepted Answer

$$0.6 	imes 0.6 = 0.36$$

Question 11

Which of the following is an example of a discrete probability distribution?

Accepted Answer

The number of cars sold by a dealership in a day.

Question 12

If $$P(A) = 0.4$$, $$P(B) = 0.5$$, and $$P(A \cap B) = 0.2$$, what is $$P(A \cup B)$$?

Accepted Answer

$$0.4 + 0.5 - 0.2 = 0.7$$

Question 13

A quality control manager wants to know if the weights of cereal boxes are normally distributed. Which graphical method should she use to evaluate normality?

Accepted Answer

Normal probability plot

Question 14

A discrete random variable $$X$$ has $$E(X) = 10$$ and $$Var(X) = 4. $$What is the standard deviation of $$X$$?

Accepted Answer

$$\sqrt{4} = 2$$

Question 15

A hospital records the number of patients arriving at the emergency room per hour. Which probability distribution is most appropriate for modeling this scenario?

Accepted Answer

Poisson distribution

Question 16

A company claims that the probability of a defective product is 0.02. If 100 products are tested, what is the expected number of defective products?

Accepted Answer

$$100 	imes 0.02 = 2$$

Question 17

Which of the following is an ethical issue related to probability in business decision-making?

Accepted Answer

Misrepresenting probabilities to influence stakeholders' decisions.

Question 18

A random variable $$X is $$normally distributed with mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What is the $$Z-$$score for $$X = 130$$?

Accepted Answer

$$Z = \frac{130 - 100}{15} = 2$$

Question 19

A sample of 25 light bulbs has a mean life of 1200 hours. The population standard deviation is 100 hours. What is the probability that a randomly selected sample of 25 bulbs has a mean life greater than 1220 hours? (Assume normality. Use $$Z = \frac{\bar{X} - \mu}{\sigma/\sqrt{n}}.)$$

Accepted Answer

$$Z = \frac{1220 - 1200}{100/5} = 1$$

Question 20

Which of the following is true about the sampling distribution of the sample mean?

Accepted Answer

It becomes approximately normal as the sample size increases, regardless of the population distribution.

Exam 2 Study Guide: Probability and Distributions in Business Statistics

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