Q: A hospital is testing a new drug. The outcome of each patient's response (improvement or no improvement) is uncertain. Which of the following best describes the 'sample space' for this experiment?

The set of all possible outcomes, $$S = \{\text{improvement}, \text{no improvement}\}$$

Q: A bag contains 3 red and 2 blue balls. If two balls are drawn without replacement, what is the probability that both are red?

$$P = \frac{3}{5} \times \frac{2}{4} = \frac{3}{10}$$

Q: A medical test for a disease is 99% accurate. If 1% of the population has the disease, what is the probability that a randomly selected person tests positive? (Use the law of total probability.)

$$P(\text{positive}) = 0.01 \times 0.99 + 0.99 \times 0.01 = 0.0198$$

Question 1

A hospital is testing a new drug. The outcome of each patient's response (improvement or no improvement) is uncertain. Which of the following best describes the 'sample space' for this experiment?

Accepted Answer

The set of all possible outcomes, $$S = \{	ext{improvement}, 	ext{no improvement}\}$$

Question 2

Suppose you roll a fair six-sided die. What is the probability of rolling an even number (event $$A$$)?

Accepted Answer

$$P(A) = \frac{3}{6} = 0.5$$

Question 3

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

Accepted Answer

$$P(A \cap B) = 0$$

Question 4

A deck contains 52 cards. How many ways can you choose 5 cards from the deck?

Accepted Answer

$$C(52,5) = \frac{52!}{5!\,47!}$$

Question 5

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

Accepted Answer

$$P(A \cap B) = P(A)P(B) = 0.2$$

Question 6

A bag contains 3 red and 2 blue balls. If two balls are drawn without replacement, what is the probability that both are red?

Accepted Answer

$$P = \frac{3}{5} 	imes \frac{2}{4} = \frac{3}{10}$$

Question 7

Which of the following is the correct formula for the complement rule?

Accepted Answer

$$P(A^c) = 1 - P(A)$$

Question 8

A medical test for a disease is 99% accurate. If 1% of the population has the disease, what is the probability that a randomly selected person tests positive? (Use the law of total probability.)

Accepted Answer

$$P(	ext{positive}) = 0.01 	imes 0.99 + 0.99 	imes 0.01 = 0.0198$$

Question 9

If $$P(A \cup B) = 0.7$$, $$P(A) = 0.5$$, and $$P(B) = 0.4$$, what is $$P(A \cap B)$$?

Accepted Answer

$$P(A \cap B) = P(A) + P(B) - P(A \cup B) = 0.2$$

Question 10

Which of the following is the correct statement of Bayes' theorem?

Accepted Answer

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

Question 11

A password consists of 4 distinct letters. How many different passwords can be formed from the 26-letter English alphabet?

Accepted Answer

$$P(26,4) = \frac{26!}{(26-4)!}$$

Question 12

If $$P(A|B) = 0.6$$ and $$P(B) = 0.5$$, what is $$P(A \cap B)$$?

Accepted Answer

$$P(A \cap B) = P(A|B)P(B) = 0.3$$

Question 13

Which of the following is NOT a property of probability according to Kolmogorov's axioms?

Accepted Answer

$$P(A) \u003c 0$$ for some event $$A$$

Question 14

A fair coin is tossed three times. What is the probability of getting exactly two heads?

Accepted Answer

$$P = C(3,2) 	imes (0.5)^2$ 	imes (0.5)^1$ = 3 	imes 0.25 	imes 0.5 = 0.375$$

Question 15

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

Accepted Answer

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

Question 16

A box contains 5 white and 3 black balls. Two balls are drawn at random without replacement. What is the probability that both are black?

Accepted Answer

$$P = \frac{3}{8} 	imes \frac{2}{7} = \frac{6}{56} = \frac{3}{28}$$

Question 17

Which of the following correctly expresses the addition rule for two events $$A$$ and $$B$$?

Accepted Answer

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

Question 18

A committee of 3 is to be formed from 5 men and 4 women. What is the number of ways to form the committee if at least one woman must be included?

Accepted Answer

$$C(9,3) - C(5,3) = 84 - 10 = 74$$

Question 19

If $$P(A) = 0.7$$ and $$P(B) = 0.2$$, what is the maximum possible value of $$P(A \cap B)$$?

Accepted Answer

$$P(A \cap B) \leq \min\{P(A), P(B)\} = 0.2$$

Question 20

A student answers a multiple-choice question with 4 options by guessing. What is the probability that the student gets the answer correct?

Accepted Answer

$$P = \frac{1}{4} = 0.25$$

Probability Fundamentals (Based on Freund's Mathematical Statistics)

Study Guide - Practice Questions

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