Q: Two dice are rolled. What is the probability that both show a 3?

$$\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$$

Q: Given the probability distribution: $$P(X=0)=0.2$$, $$P(X=1)=0.5$$, $$P(X=2)=0.3$$, what is the mean $$\mu of$$ $$X$$?

$$\mu = 0 \times 0.2 + 1 \times 0.5 + 2 \times 0.3 = 1.1$$

Q: A binomial experiment has $$n=12$$ trials and probability of success $$p=0.25. $$What is the standard deviation $$\sigma$$?

$$\sigma = \sqrt{12 \times 0.25 \times 0.75} \approx 1.5$$

Question 1

A bag contains 7 green and 5 yellow marbles. If one marble is drawn at random, what is the probability that it is yellow?

Accepted Answer

$$\frac{5}{12}$$

Question 2

A basketball player makes 75 out of 100 free throws. What is the empirical probability that the player makes a free throw?

Accepted Answer

$$0.75$$

Question 3

If $$P(A) = 0.40$$, what is the probability of the complement event $$A'$$?

Accepted Answer

$$0.60$$

Question 4

A card is drawn from a standard deck. What is the probability of drawing a queen or a spade?

Accepted Answer

$$\frac{4}{52} + \frac{13}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}$$

Question 5

Two dice are rolled. What is the probability that both show a 3?

Accepted Answer

$$\frac{1}{6} 	imes \frac{1}{6} = \frac{1}{36}$$

Question 6

In a class, 20 students are left-handed, and 8 of them play a musical instrument. What is the probability that a randomly selected left-handed student plays an instrument?

Accepted Answer

$$\frac{8}{20} = 0.4$$

Question 7

Which statement is TRUE about mutually exclusive and independent events?

Accepted Answer

Mutually exclusive events with positive probabilities are not independent.

Question 8

A fair coin is flipped 4 times. What is the probability of getting at least one head?

Accepted Answer

$$1 - (0.5)^4$ = 1 - 0.0625 = 0.9375$$

Question 9

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

Accepted Answer

P(X=0)=0.2, P(X=1)=0.5, P(X=2)=0.3

Question 10

Given the probability distribution: $$P(X=0)=0.2$$, $$P(X=1)=0.5$$, $$P(X=2)=0.3$$, what is the mean $$\mu of$$ $$X$$?

Accepted Answer

$$\mu = 0 	imes 0.2 + 1 	imes 0.5 + 2 	imes 0.3 = 1.1$$

Question 11

A binomial experiment has $$n=12$$ trials and probability of success $$p=0.25. $$What is the standard deviation $$\sigma$$?

Accepted Answer

$$\sigma = \sqrt{12 	imes 0.25 	imes 0.75} \approx 1.5$$

Question 12

In a binomial setting with $$n=8$$ and $$p=0.6$$, what is the probability of getting at least 6 successes?

Accepted Answer

$$1 - P(X \leq 5)$$

Question 13

A normal distribution has $$\mu=50$$ and $$\sigma=8. $$What percentage of values are expected to fall between 42 and 58?

Accepted Answer

About 68%

Question 14

For a normal distribution with $$\mu=100$$ and $$\sigma=20$$, what is the $$z-$$score for $$x=140$$?

Accepted Answer

$$z = \frac{140 - 100}{20} = 2$$

Question 15

If $$X \sim N(70, 5)$$, what is the probability that $$X \u003c 75$$?

Accepted Answer

$$\Phi\left(\frac{75-70}{5}\right) = \Phi(1) \approx 0.8413$$

Question 16

A sample of $$n=36 is $$drawn from a population with $$\mu=80$$ and $$\sigma=18. $$What is the standard error (SE) of the sample mean?

Accepted Answer

$$SE = \frac{18}{\sqrt{36}} = 3$$

Question 17

A population proportion is $$p=0.4$$ and a random sample of $$n=100 is $$taken. What is the standard error (SE) for the sample proportion?

Accepted Answer

$$SE = \sqrt{\frac{0.4 	imes 0.6}{100}} \approx 0.049$$

Question 18

Which condition must be met for the Central Limit Theorem to apply to the sampling distribution of the sample mean when the population is not normal?

Accepted Answer

Sample size $$n$$ must be sufficiently large (often $$n \geq 30)$$

Question 19

A histogram of exam scores is bell-shaped and symmetric with one main peak and no severe outliers. Which model is most appropriate?

Accepted Answer

Normal distribution

Statistics Exam 2 Study Guide: Probability, Discrete Random Variables, and Normal Distributions

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