Q: A bag contains 5 red, 3 blue, and 2 green marbles. If one marble is drawn at random, what is the probability it is blue?

$$P(\text{blue}) = \frac{3}{10}$$

Q: A survey finds that 60% of people like coffee, 50% like tea, and 30% like both. What is the probability that a randomly selected person likes coffee or tea?

$$P(\text{coffee or tea}) = 0.60 + 0.50 - 0.30 = 0.80$$

Q: If $$P(A) = 0.4$$ and $$P(B) = 0.5$$, and A and B are independent, what is $$P(A \text{ and } B)$$?

$$P(A \text{ and } B) = 0.4 \times 0.5 = 0.2$$

Q: A password consists of 3 letters followed by 2 digits. If letters and digits can be repeated, how many different passwords are possible?

$$26^3$$ \times $$10^2 = 175$$,760$$

Q: Given the following probability distribution for $$X$$: $$P(X=0)=0.2$$, $$P(X=1)=0.5$$, $$P(X=2)=0.3. $$What is the mean (expected value) of $$X$$?

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

Question 1

A pharmaceutical company tests a new drug on 10,000 patients. The proportion of patients who experience a side effect is 0.12. According to the Law of Large Numbers, what will happen if the company tests the drug on 1,000,000 patients?

Accepted Answer

The proportion of patients experiencing the side effect will get closer to the true probability of 0.12.

Question 2

Which of the following sets of probabilities represents a valid probability model?

Accepted Answer

$$P(A) = 0.3$$, $$P(B) = 0.5$$, $$P(C) = 0.2$$

Question 3

A weather forecaster says, 'Based on my experience, there is a 70% chance of rain tomorrow.' What type of probability is this?

Accepted Answer

Subjective probability

Question 4

A bag contains 5 red, 3 blue, and 2 green marbles. If one marble is drawn at random, what is the probability it is blue?

Accepted Answer

$$P(	ext{blue}) = \frac{3}{10}$$

Question 5

Which of the following pairs of events are disjoint (mutually exclusive)?

Accepted Answer

Rolling a die: Event A = 'rolling a 2', Event B = 'rolling a 5'

Question 6

A survey finds that 60% of people like coffee, 50% like tea, and 30% like both. What is the probability that a randomly selected person likes coffee or tea?

Accepted Answer

$$P(	ext{coffee or tea}) = 0.60 + 0.50 - 0.30 = 0.80$$

Question 7

If $$P(A) = 0.4$$ and $$P(B) = 0.5$$, and A and B are independent, what is $$P(A 	ext{ and } B)$$?

Accepted Answer

$$P(A 	ext{ and } B) = 0.4 	imes 0.5 = 0.2$$

Question 8

A password consists of 3 letters followed by 2 digits. If letters and digits can be repeated, how many different passwords are possible?

Accepted Answer

$$26^3$$ 	imes $$10^2 = 175$$,760$$

Question 9

Which of the following is a discrete random variable?

Accepted Answer

The number of defective light bulbs in a box

Question 10

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

Accepted Answer

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

Question 11

Which of the following is NOT a property of a normal density curve?

Accepted Answer

The total area under the curve is less than 1.

Question 12

If $$X is $$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 13

What is the probability of observing any single specific value of a continuous random variable?

Accepted Answer

0

Question 14

Suppose the area to the left of $$z = 1.25$$ under the standard normal curve is 0.8944. What is the area to the right of $$z = 1.25$$?

Accepted Answer

$$1 - 0.8944 = 0.1056$$

Question 15

A lottery ticket costs $$2 and pays 100 if $$you win (probability 0.001), otherwise you win nothing. What is the expected value of buying one ticket?

Accepted Answer

$$E = (100 - 2) 	imes 0.001 + (-2) 	imes 0.999 = -1.902$$

Question 16

If the area to the left of a $$z-$$score is 0.9750, what is the corresponding $$z-$$score (rounded to two decimals)?

Accepted Answer

$$z = 1.96$$

Question 17

A committee of 4 is to be chosen from 10 people. How many different committees are possible?

Accepted Answer

$$\binom{10}{4} = 210$$

Question 18

A factory produces light bulbs with a 2% defect rate. If you randomly select 5 bulbs, what is the probability that at least one is defective?

Accepted Answer

$$1 - (0.98)^5$ \approx 0.0961$$

Question 19

Given $$P(A|B) = 0.3$$ and $$P(B) = 0.5$$, what is $$P(A 	ext{ and } B)$$?

Accepted Answer

$$P(A 	ext{ and } B) = P(A|B) 	imes P(B) = 0.3 	imes 0.5 = 0.15$$

Probability, Discrete and Continuous Probability Distributions, and the Normal Distribution: Study Guide

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