Q: If two events are independent, which of the following is true?

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

Q: Suppose $$P(A) = 0.20$$, $$P(B) = 0.40$$, and $$P(A \cap B) = 0.08. $$Are events A and B independent?

Yes, because $$0.20 \times 0.40 = 0.08$$

Question 1

Suppose you applied for two grants: a government grant (G) and a private grant (P). The probability of receiving the government grant is $$P(G) = 0.5$$, the probability of receiving the private grant is $$P(P) = 0.45$$, and the probability of receiving both is $$P(G \cap P) = 0.2. $$What is the probability that you receive at least one grant?

Accepted Answer

$$0.75$$

Question 2

A recent survey at a college found that the probability of a student being a freshman is $$P(F) = 0.60$$, the probability of being a business major is $$P(B) = 0.40$$, and the probability of being both is $$P(F \cap B) = 0.25. $$What is the probability that a randomly selected student is either a freshman or a business major?

Accepted Answer

$$0.80$$

Question 3

Given $$P(A) = 0.80$$, $$P(B) = 0.60$$, and $$P(A \cap B) = 0.48$$, what is the probability that either event A or event B occurs?

Accepted Answer

$$0.92$$

Question 4

If two events are independent, which of the following is true?

Accepted Answer

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

Question 5

Suppose $$P(A) = 0.20$$, $$P(B) = 0.40$$, and $$P(A \cap B) = 0.08. $$Are events A and B independent?

Accepted Answer

Yes, because $$0.20 	imes 0.40 = 0.08$$

Question 6

Which of the following best describes mutually exclusive events?

Accepted Answer

They cannot occur at the same time, so $$P(A \cap B) = 0$$

Question 7

A contingency table shows that 20 out of 500 patients have both flu symptoms and high cholesterol. What is the probability that a randomly selected patient has both conditions?

Accepted Answer

$$0.04$$

Question 8

In a study, 80 out of 500 patients have high cholesterol and do not exhibit flu symptoms. What is the probability that a randomly selected patient falls into this category?

Accepted Answer

$$0.16$$

Question 9

Given a normal distribution with mean $$432$$ and standard deviation $$17$$, what percentage of observations fall between $$399$$ and $$447$$?

Accepted Answer

$$83.85\%$$

Question 10

If $$P(A) = 0.5$$, $$P(B) = 0.45$$, and $$P(A \cap B) = 0.2$$, what is $$P(A \cup B)$$?

Accepted Answer

$$0.75$$

Question 11

Which of the following is NOT a property of probability?

Accepted Answer

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

Question 12

If two events are not mutually exclusive, which of the following statements is true?

Accepted Answer

They can occur at the same time

Question 13

A new apartment building is under construction. Potential tenants have been polled about their preferences for a pool (event A) and a gym (event B). If $$P(A) = 0.45$$, $$P(B) = 0.27$$, and $$P(A \cap B) = 0.12$$, what is the probability that a tenant wants either a pool or a gym?

Accepted Answer

$$0.60$$

Question 14

If $$P(A) = 0.20$$, $$P(B) = 0.40$$, and $$P(A \cap B) = 0.08$$, what is $$P(A \cup B)$$?

Accepted Answer

$$0.52$$

Question 15

Which of the following best describes the empirical rule for a normal distribution?

Accepted Answer

About $$68\% of $$data falls within one standard deviation of the mean

Question 16

If $$P(A) = 0.5$$ and $$P(B) = 0.45$$, and A and B are independent, what is $$P(A \cap B)$$?

Accepted Answer

$$0.225$$

Probability & Contingency Tables: Step-by-Step Statistics Study Guide

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