Q: Suppose 80 out of 400 students can both raise one eyebrow and roll their tongue. What is the probability that a randomly selected student can do both?

$$P(\text{Both}) = \frac{80}{400}$$

Q: A discrete random variable $$X$$ has the following probability distribution: $$P(X=1)=0.2$$, $$P(X=2)=0.5$$, $$P(X=3)=0.3. $$What is the expected value $$E(X)$$?

$$E(X) = 1 \times 0.2 + 2 \times 0.5 + 3 \times 0.3 = 2.1$$

Q: If $$X is a $$binomial random variable with $$n=20$$ and $$p=0.3$$, what is the standard deviation of $$X$$?

$$\sigma = \sqrt{20 \times 0.3 \times 0.7}$$

Q: If $$A$$ and $$B$$ are independent events, which of the following is true?

$$P(A \cap B) = P(A) \times P(B)$$

Q: A game involves spinning a wheel with 5 equal sections labeled 1 through 5. What is the probability of spinning a 3 or a 5 in one spin?

$$P(3 \text{ or } 5) = \frac{2}{5}$$

Question 1

A company claims that 51% of its employees are women. In a random sample of 10 employees, what is the probability that exactly 6 are women? Use the binomial probability formula.

Accepted Answer

$$P(X=6) = \binom{10}{6} (0.51)^6$ (0.49)^4$

Question 2

Which of the following is the correct formula for the probability of the union of two events $$A$$ and $$B$$?

Accepted Answer

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

Question 3

Suppose 80 out of 400 students can both raise one eyebrow and roll their tongue. What is the probability that a randomly selected student can do both?

Accepted Answer

$$P(	ext{Both}) = \frac{80}{400}$$

Question 4

If $$P(A) = 0.3$$, $$P(B) = 0.5$$, and $$P(A \cap B) = 0.1$$, what is $$P(A|B)$$?

Accepted Answer

$$P(A|B) = \frac{0.1}{0.5} = 0.2$$

Question 5

A discrete random variable $$X$$ has the following probability distribution: $$P(X=1)=0.2$$, $$P(X=2)=0.5$$, $$P(X=3)=0.3. $$What is the expected value $$E(X)$$?

Accepted Answer

$$E(X) = 1 	imes 0.2 + 2 	imes 0.5 + 3 	imes 0.3 = 2.1$$

Question 6

Which of the following is a condition for a binomial setting?

Accepted Answer

Each trial is independent.

Question 7

If $$X is a $$binomial random variable with $$n=20$$ and $$p=0.3$$, what is the standard deviation of $$X$$?

Accepted Answer

$$\sigma = \sqrt{20 	imes 0.3 	imes 0.7}$$

Question 8

A continuous random variable $$Y is $$uniformly distributed between 0 and 10. What is $$P(2 \u003c Y \u003c 5)$$?

Accepted Answer

$$P(2 \u003c Y \u003c 5) = \frac{5-2}{10-0} = 0.3$$

Question 9

If $$A$$ and $$B$$ are independent events, which of the following is true?

Accepted Answer

$$P(A \cap B) = P(A) 	imes P(B)$$

Question 10

A game involves spinning a wheel with 5 equal sections labeled 1 through 5. What is the probability of spinning a 3 or a 5 in one spin?

Accepted Answer

$$P(3 	ext{ or } 5) = \frac{2}{5}$$

Question 11

Suppose $$X$$ and $$Y$$ are independent random variables with $$E(X) = 10$$, $$E(Y) = 15$$, $$Var(X) = 4$$, and $$Var(Y) = 9. $$What is $$E(X+Y)$$ and $$Var(X+Y)$$?

Accepted Answer

$$E(X+Y) = 25$$, $$Var(X+Y) = 13$$

Question 12

A student takes a 5-question multiple-choice quiz with 4 answer choices per question, guessing on each question. What is the probability the student gets exactly 2 questions correct?

Accepted Answer

$$P(X=2) = \binom{5}{2} (0.25)^2$ (0.75)^3$

Question 13

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

Accepted Answer

$$P(A \cup B) = 0.4 + 0.5 - 0.2 = 0.7$$

Question 14

A random variable $$X$$ has mean $$\mu = 8$$ and standard deviation $$\sigma = 2. If$$ $$Y = 3X + 5$$, what is the mean and standard deviation of $$Y$$?

Accepted Answer

$$E(Y) = 29$$, $$SD(Y) = 6$$

Question 15

Which of the following is the formula for the variance of the sum of two independent random variables $$X$$ and $$Y$$?

Accepted Answer

$$Var(X+Y) = Var(X) + Var(Y)$$

Question 16

A geometric random variable $$Y$$ has probability of success $$p=0.2. $$What is the probability that the first success occurs on the 4th trial?

Accepted Answer

$$P(Y=4) = (1-0.2$$)^{3}$$ 	imes 0.2$$

Question 17

If a random variable $$X is $$normally distributed with mean $$\mu = 100$$ and standard deviation $$\sigma = 15$$, what is $$P(X \u003e 130)$$?

Accepted Answer

$$P(X \u003e 130) = P(Z \u003e 2) \approx 0.0228$$

Question 18

A hospital records the number of patients arriving each hour as a discrete random variable. Which of the following is NOT a property of a probability distribution for this variable?

Accepted Answer

The sum of all probabilities is greater than 1.

Probability and Random Variables: Core Concepts and Applications

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