Q: A fair six-sided die is rolled. What is the probability of rolling either a 2 or a 5?

$$P(2 \text{ or } 5) = \frac{2}{6} = 0.333$$

Q: A bag contains 5 red, 3 blue, and 2 green marbles. If one marble is drawn at random, what is the probability that it is not blue?

$$P(\text{not blue}) = 1 - P(\text{blue}) = 1 - \frac{3}{10} = 0.7$$

Q: In a lottery, the probability of winning is 0.001. What are the actual odds against winning?

$$\text{Odds against} = \frac{P(\text{not win})}{P(\text{win})} = \frac{0.999}{0.001} = 999:1$$

Q: A password consists of 3 letters (A-Z). How many different passwords are possible if letters can be repeated?

$$26 \times 26 \times 26 = 17,576$$

Q: A deck of cards has 52 cards. What is the probability of drawing an ace or a king in a single draw?

$$P(\text{ace or king}) = P(\text{ace}) + P(\text{king}) = \frac{4}{52} + \frac{4}{52} = \frac{8}{52} = 0.154$$

Q: If two events A and B are independent, which formula correctly gives the probability that both occur?

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

Question 1

A medical researcher observes that out of 200 patients, 30 develop a certain side effect after taking a new drug. Using the relative frequency approximation, what is the probability that a randomly selected patient will develop the side effect?

Accepted Answer

$$P(A) = \frac{30}{200} = 0.15$$

Question 2

A fair six-sided die is rolled. What is the probability of rolling either a 2 or a 5?

Accepted Answer

$$P(2 	ext{ or } 5) = \frac{2}{6} = 0.333$$

Question 3

Which of the following best describes the law of large numbers?

Accepted Answer

As the number of trials increases, the relative frequency probability approaches the actual 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 that it is not blue?

Accepted Answer

$$P(	ext{not blue}) = 1 - P(	ext{blue}) = 1 - \frac{3}{10} = 0.7$$

Question 5

In a lottery, the probability of winning is 0.001. What are the actual odds against winning?

Accepted Answer

$$	ext{Odds against} = \frac{P(	ext{not win})}{P(	ext{win})} = \frac{0.999}{0.001} = 999:1$$

Question 6

A student flips a coin 10 times and gets heads every time. According to the law of large numbers, what can be said about the probability of getting heads on the next flip?

Accepted Answer

The probability remains $$0.5$$; previous outcomes do not affect future flips.

Question 7

A password consists of 3 letters (A-Z). How many different passwords are possible if letters can be repeated?

Accepted Answer

$$26 	imes 26 	imes 26 = 17,576$$

Question 8

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

Accepted Answer

$$_{10}C_4$ = \frac{10!}{(10-4)!4!} = 210$$

Question 9

A deck of cards has 52 cards. What is the probability of drawing an ace or a king in a single draw?

Accepted Answer

$$P(	ext{ace or king}) = P(	ext{ace}) + P(	ext{king}) = \frac{4}{52} + \frac{4}{52} = \frac{8}{52} = 0.154$$

Question 10

If two events A and B are independent, which formula correctly gives the probability that both occur?

Accepted Answer

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

Question 11

A factory has two machines. The probability that machine A works is 0.9, and the probability that machine B works is 0.8. Assuming independence, what is the probability that both machines work?

Accepted Answer

$$P(A 	ext{ and } B) = 0.9 	imes 0.8 = 0.72$$

Question 12

A box contains 3 red and 2 blue balls. Two balls are drawn without replacement. What is the probability that both are red?

Accepted Answer

$$P(	ext{both red}) = \frac{3}{5} 	imes \frac{2}{4} = 0.3$$

Question 13

A student randomly guesses on a 5-question true/false quiz. What is the probability that the student gets at least one question correct?

Accepted Answer

$$P(	ext{at least one correct}) = 1 - P(	ext{none correct}) = 1 - (0.5)^5$ = 1 - 0.03125 = 0.96875$$

Question 14

Which of the following statements about mutually exclusive (disjoint) events is correct?

Accepted Answer

Mutually exclusive events cannot occur at the same time.

Question 15

A bag contains 4 white and 6 black balls. Two balls are drawn with replacement. What is the probability that both balls are white?

Accepted Answer

$$P(	ext{both white}) = \frac{4}{10} 	imes \frac{4}{10} = 0.16$$

Question 16

A password consists of 4 digits (0-9), but no digit can be repeated. How many different passwords are possible?

Accepted Answer

$$10 	imes 9 	imes 8 	imes 7 = 5040$$

Question 17

A jar contains 5 identical red candies, 3 identical blue candies, and 2 identical green candies. How many different ways can all 10 candies be arranged in a row?

Accepted Answer

$$\frac{10!}{5!3!2!} = 2520$$

Question 18

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

Accepted Answer

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

Question 19

A researcher wants to estimate the probability of a rare disease in a population using a simulation. Which approach is most appropriate?

Accepted Answer

Simulation approach, because the classical and relative frequency methods may not be feasible.

Probability and Counting Rules in Introductory Statistics

Guida di studio - Domande di pratica

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