Q: A company is considering three possible offers for a property: $$450,000$$ with probability $$0.45$$, $$520,000$$ with probability $$0.15$$, $$580,000$$ with probability $$0.25$$, and $$640,000$$ with probability $$0.15. $$What is the expected value of the offer? Use the formula: $$ E(X) = \sum [x \cdot P(x)] $$

$$E(X) = 450,000 \times 0.45 + 520,000 \times 0.15 + 580,000 \times 0.25 + 640,000 \times 0.15 = 202,500 + 78,000 + 145,000 + 96,000 = 521,500$$

Q: A binomial experiment consists of $$n = 5$$ trials with probability of success $$p = 0.6. $$What is the probability of getting exactly 3 successes? Use the formula: $$ P(x) = {n \choose x} p^x (1-p$$)^{n-x}$$ $$

$$P(3) = {5 \choose 3} (0.6)^3$ (0.4)^2$ = 10 \times 0.216 \times 0.16 = 0.3456$$

Question 1

Which of the following is a discrete random variable?

Accepted Answer

The number of cars passing through an intersection in one hour

Question 2

Given the probability distribution for the profit $$X$$ from a $$1 bet in $$Chuck-a-Luck:

Number of Dice Matching | Profit | Probability
0 | $$-1$$ | $$0.5787
1$$ | $$1$$ | $$0.3472
2$$ | $$2$$ | $$0.0695
3$$ | $$3$$ | $$0.0046$$

What is the mean (expected value) of $$X$$?

Use the formula:
$$
\mu_X = \sum [x \cdot P(x)]
$$

Accepted Answer

$$-0.09$$

Question 3

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

Accepted Answer

The probability of success changes from trial to trial

Question 4

Suppose a binomial experiment has $$n = 20$$ trials and probability of success $$p = 0.3. $$What is the mean and standard deviation of the number of successes?

Use the formulas:
$$
\mu = np
$$
$$
\sigma = \sqrt{np(1-p)}
$$

Accepted Answer

$$\mu = 6$$, $$\sigma \approx 2.05$$

Question 5

Which of the following statements about the probability distribution of a discrete random variable is TRUE?

Accepted Answer

Each probability must be between 0 and 1, inclusive

Question 6

A random variable $$X$$ has the following probability distribution:

$$x$$ | $$P(x)$$
---|------
2 | 0.2
4 | 0.5
6 | 0.3

What is the standard deviation of $$X$$?

Use the formulas:
$$
\mu = \sum [x \cdot P(x)]
$$
$$
\sigma = \sqrt{\sum [(x - \mu)^2$ \cdot P(x)]}
$$

Accepted Answer

$$\mu = 4.6$$, $$\sigma \approx 1.62$$

Question 7

In the context of binomial probability, which mathematical symbol represents 'at least'?

Accepted Answer

$$\geq$$

Question 8

A company is considering three possible offers for a property: $$450,000$$ with probability $$0.45$$, $$520,000$$ with probability $$0.15$$, $$580,000$$ with probability $$0.25$$, and $$640,000$$ with probability $$0.15. $$What is the expected value of the offer?

Use the formula:
$$
E(X) = \sum [x \cdot P(x)]
$$

Accepted Answer

$$E(X) = 450,000 	imes 0.45 + 520,000 	imes 0.15 + 580,000 	imes 0.25 + 640,000 	imes 0.15 = 202,500 + 78,000 + 145,000 + 96,000 = 521,500$$

Question 9

Which scenario below describes a continuous random variable?

Accepted Answer

The time (in seconds) it takes for a runner to finish a race

Question 10

Suppose you roll three dice and bet $$1 on a $$number in Chuck-a-Luck. What is the probability that exactly two dice match your chosen number?

Use the probability distribution:

Number of Dice Matching | Probability
2 | $$0.0695$$

Accepted Answer

$$0.0695$$

Question 11

Which formula gives the standard deviation of a discrete random variable $$X$$ with mean $$\mu$$?

Accepted Answer

$$\sigma_X = \sqrt{\sum [(x - \mu)^2$ \cdot P(x)]}$$

Question 12

A binomial experiment consists of $$n = 5$$ trials with probability of success $$p = 0.6. $$What is the probability of getting exactly 3 successes?

Use the formula:
$$
P(x) = {n \choose x} p^x (1-p$$)^{n-x}$$
$$

Accepted Answer

$$P(3) = {5 \choose 3} (0.6)^3$ (0.4)^2$ = 10 	imes 0.216 	imes 0.16 = 0.3456$$

Question 13

Which of the following is a valid probability distribution for a discrete random variable?

Accepted Answer

$$P(1) = 0.3$$, $$P(2) = 0.5$$, $$P(3) = 0.2$$

Question 14

In a binomial experiment with $$n = 10$$ and $$p = 0.2$$, what is the probability of getting at least 2 successes?

Express your answer using binomial probabilities.

Accepted Answer

$$P(x \geq 2) = 1 - [P(0) + P(1)]$$

Question 15

Which of the following best describes the interpretation of the mean of a discrete random variable?

Accepted Answer

It is the long-run average value expected after many repetitions of the experiment

Question 16

A probability experiment consists of rolling a die 4 times. Let $$X be $$the number of times a 6 appears. Which of the following statements is TRUE?

Accepted Answer

$$X is a $$discrete random variable with possible values 0, 1, 2, 3, 4

Question 17

Which formula gives the probability of obtaining $$x$$ successes in $$n$$ independent binomial trials with probability $$p of $$success?

Accepted Answer

$$P(x) = {n \choose x} p^x (1-p$$)^{n-x}$$

Question 18

A random variable $$X$$ has mean $$\mu = 5$$ and standard deviation $$\sigma = 2. If $$you repeat the experiment $$n$$ times and calculate the average $$\bar{x}$$, what happens to $$|\bar{x} - \mu| as$$ $$n$$ increases?

Accepted Answer

It gets closer to zero

Discrete and Binomial Probability Distributions: Key Concepts and Applications

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