Q: A fair die is rolled 4 times. What is the probability of getting exactly 2 sixes? (Use $$p = \frac{1}{6}$$, $$n = 4$$, $$x = 2)$$

$$P(X=2) = 6 \times \left(\frac{1}{6}\$$right)^2$$ \times \left(\frac{5}{6}\$$right)^2$$

Q: The standard deviation of a binomial random variable with $$n = 20$$ and $$p = 0.4 is$$:

$$\sigma = \sqrt{20 \times 0.4 \times 0.6}$$

Question 1

Which of the following is a discrete random variable?

Accepted Answer

The number of red cars passing through an intersection in an hour

Question 2

A probability distribution for a discrete random variable X must satisfy which of the following conditions?

Accepted Answer

$$\sum P(X) = 1$$

Question 3

Suppose a random variable X represents the number of heads in three coin tosses. What is the probability that X = 2?

Accepted Answer

$$P(X=2) = \frac{3}{8}$$

Question 4

Given the following discrete probability distribution for X: X: 0, 1, 2; P(X): 0.2, 0.5, 0.3. What is the mean of X?

Accepted Answer

$$\mu = 1.1$$

Question 5

In the Chuck-A-Luck game, the probability of losing $$1 is 0.5787. What is the expected value (mean) of the profit X, given the following probabilities: -1$$: 0.5787$$, 1$$: 0.3472$$, 2$$: 0.0695$$, 3$$: 0.0046$?

Accepted Answer

$$E(X) = -0.11$$

Question 6

Which of the following scenarios describes a binomial experiment?

Accepted Answer

Counting the number of defective bulbs in a sample of 20, where each bulb is either defective or not

Question 7

If a binomial experiment has $$n = 10$$ trials and probability of success $$p = 0.7$$, what is the mean number of successes?

Accepted Answer

$$\mu = 7$$

Question 8

A fair die is rolled 4 times. What is the probability of getting exactly 2 sixes? (Use $$p = \frac{1}{6}$$, $$n = 4$$, $$x = 2)$$

Accepted Answer

$$P(X=2) = 6 	imes \left(\frac{1}{6}\$$right)^2$$ 	imes \left(\frac{5}{6}\$$right)^2$$

Question 9

The standard deviation of a binomial random variable with $$n = 20$$ and $$p = 0.4 is$$:

Accepted Answer

$$\sigma = \sqrt{20 	imes 0.4 	imes 0.6}$$

Question 10

A medical test has a 95% chance of correctly identifying a disease. If 3 patients are tested, what is the probability that all 3 are correctly identified?

Accepted Answer

$$P = 0.95^3$

Question 11

Which of the following phrases corresponds to the mathematical symbol $$\leq$$?

Accepted Answer

"no more than"

Question 12

Suppose the probability that a flight is on time is 0.7. If 4 flights are randomly selected, what is the probability that exactly 3 are on time?

Accepted Answer

$$P(X=3) = \binom{4}{3} (0.7)^3$ (0.3)^1$

Question 13

If the mean profit for an insurance policy is $$E(X) = 86.14$$, what does this mean in the context of the insurance company?

Accepted Answer

On average, the company expects to earn $86.14 per policy in the long run.

Question 14

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

Accepted Answer

Each trial must have more than two possible outcomes.

Question 15

A student randomly guesses on a 5-question true/false quiz. What is the probability the student gets exactly 3 questions correct?

Accepted Answer

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

Question 16

If a discrete random variable X has possible values 1, 2, 3 with probabilities 0.2, 0.5, and 0.3, what is the variance of X?

Accepted Answer

$$Var(X) = 0.49$$

Discrete and Binomial Probability Distributions: Study Notes

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