Q: If $$P(A) = 0.3$$ and $$P(B) = 0.5$$, and A and B are mutually exclusive, what is $$P(A \text{ or } B)$$?

$$P(A \text{ or } B) = P(A) + P(B) = 0.8$$

Q: Given $$P(E) = 0.7$$, what is the probability that event E does not occur?

$$P(\text{not } E) = 1 - P(E) = 0.3$$

Q: If $$P(A) = 0.4$$, $$P(B) = 0.5$$, and $$P(A \text{ and } B) = 0.2$$, what is $$P(A \text{ or } B)$$?

$$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) = 0.4 + 0.5 - 0.2 = 0.7$$

Q: If $$P(A) = 0.6$$, $$P(B|A) = 0.5$$, what is $$P(A \text{ and } B)$$?

$$P(A \text{ and } B) = P(A) \times P(B|A) = 0.6 \times 0.5 = 0.3$$

Q: A medical test for a disease is positive in 95% of cases when the disease is present and in 5% of cases when the disease is absent. If 1% of the population has the disease, what is the probability that a randomly selected person who tests positive actually has the disease? (Use Bayes’s Rule.)

$$P(\text{Disease} | \text{Positive}) = \frac{0.95 \times 0.01}{0.95 \times 0.01 + 0.05 \times 0.99} \approx 0.161$$

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

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

Question 1

A fair coin is tossed 100 times. Which property of probability is illustrated by the proportion of heads approaching 0.5 as the number of tosses increases?

Accepted Answer

The Law of Large Numbers, which states that the relative frequency of an event approaches its theoretical probability as the number of trials increases.

Question 2

Suppose you roll two fair six-sided dice. What is the probability that the sum of the dice is 7?

Accepted Answer

There are 6 ways to get a sum of 7 out of 36 possible outcomes, so $$\frac{6}{36} = \frac{1}{6}.$$

Question 3

Which of the following best describes mutually exclusive events?

Accepted Answer

Events that cannot occur at the same time and have no outcomes in common.

Question 4

If $$P(A) = 0.3$$ and $$P(B) = 0.5$$, and A and B are mutually exclusive, what is $$P(A 	ext{ or } B)$$?

Accepted Answer

$$P(A 	ext{ or } B) = P(A) + P(B) = 0.8$$

Question 5

A university faculty table shows that 0.045 of all faculty are full professors under age 40. What type of probability is this?

Accepted Answer

Joint probability

Question 6

Given $$P(E) = 0.7$$, what is the probability that event E does not occur?

Accepted Answer

$$P(	ext{not } E) = 1 - P(E) = 0.3$$

Question 7

If $$P(A) = 0.4$$, $$P(B) = 0.5$$, and $$P(A 	ext{ and } B) = 0.2$$, what is $$P(A 	ext{ or } B)$$?

Accepted Answer

$$P(A 	ext{ or } B) = P(A) + P(B) - P(A 	ext{ and } B) = 0.4 + 0.5 - 0.2 = 0.7$$

Question 8

In a contingency table, what is a marginal probability?

Accepted Answer

The probability of a single event occurring, found by summing the joint probabilities for that event.

Question 9

If $$P(A) = 0.6$$, $$P(B|A) = 0.5$$, what is $$P(A 	ext{ and } B)$$?

Accepted Answer

$$P(A 	ext{ and } B) = P(A) 	imes P(B|A) = 0.6 	imes 0.5 = 0.3$$

Question 10

A medical test for a disease is positive in 95% of cases when the disease is present and in 5% of cases when the disease is absent. If 1% of the population has the disease, what is the probability that a randomly selected person who tests positive actually has the disease? (Use Bayes’s Rule.)

Accepted Answer

$$P(	ext{Disease} | 	ext{Positive}) = \frac{0.95 	imes 0.01}{0.95 	imes 0.01 + 0.05 	imes 0.99} \approx 0.161$$

Question 11

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

Accepted Answer

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

Question 12

A box contains 6 defective and 94 nondefective TVs. If 5 TVs are selected at random, what is the number of ways exactly 2 defective TVs can be chosen?

Accepted Answer

$$\binom{6}{2} 	imes \binom{94}{3}$$

Question 13

How many possible permutations are there of 3 letters chosen from a collection of 5 distinct letters?

Accepted Answer

$$\frac{5!}{(5-3)!} = 60$$

Question 14

What is the difference between a permutation and a combination?

Accepted Answer

A permutation considers the order of objects, while a combination does not.

Question 15

If a sample space consists of N equally likely outcomes, and an event E can occur in f ways, what is the probability of E?

Accepted Answer

$$P(E) = \frac{f}{N}$$

Question 16

A contingency table shows that the probability a randomly selected faculty member is an associate professor is 0.331. What type of probability is this?

Accepted Answer

Marginal probability

Question 17

If $$P(A) = 0.2$$, $$P(B) = 0.3$$, and $$P(A 	ext{ and } B) = 0.06$$, are A and B independent?

Accepted Answer

Yes, because $$P(A 	ext{ and } B) = P(A) 	imes P(B) = 0.2 	imes 0.3 = 0.06$$

Question 18

How many possible samples of size n can be drawn from a population of size N?

Accepted Answer

$$\binom{N}{n}$$

Probability Concepts – Study Notes

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