Q: A student flips a coin 10 times and gets heads 7 times. What is the relative frequency approximation for the probability of getting heads?

$$P(\text{heads}) = \frac{7}{10} = 0.7$$

Q: If the probability of passing a statistics test is 0.6, what is the probability of failing the test?

$$P(\text{fail}) = 1 - 0.6 = 0.4$$

Q: A deck of cards contains 52 cards. What is the probability of drawing either a heart or a king from the deck? (There are 13 hearts and 4 kings, with 1 card being both a heart and a king.)

$$P(A \text{ or } B) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}$$

Q: A factory has two machines. The probability that machine A works is 0.9, and the probability that machine B works given that A works is 0.8. What is the probability that both machines work?

$$P(A \text{ and } B) = 0.9 \times 0.8 = 0.72$$

Q: In a lottery, the probability of winning is 0.01. What are the actual odds against winning?

$$\text{Odds against} = \frac{P(\text{not win})}{P(\text{win})} = \frac{0.99}{0.01} = 99:1$$

Q: A bag contains 5 red balls and 3 blue balls. If two balls are drawn without replacement, what is the probability that both are red?

$$P(\text{both red}) = \frac{5}{8} \times \frac{4}{7} = \frac{20}{56} = \frac{5}{14}$$

Q: A password consists of 3 letters, each chosen from the 26 English letters. How many different passwords are possible if letters can be repeated?

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

Question 1

A medical researcher conducts an experiment to estimate the probability that a new drug cures a disease. Out of 200 patients treated, 160 are cured. Using the relative frequency approximation, what is the probability that the drug cures the disease?

Accepted Answer

$$P(A) = \frac{160}{200} = 0.8$$

Question 2

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

Accepted Answer

$$P(A) = \frac{2}{6} = \frac{1}{3}$$

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 student flips a coin 10 times and gets heads 7 times. What is the relative frequency approximation for the probability of getting heads?

Accepted Answer

$$P(	ext{heads}) = \frac{7}{10} = 0.7$$

Question 5

If the probability of passing a statistics test is 0.6, what is the probability of failing the test?

Accepted Answer

$$P(	ext{fail}) = 1 - 0.6 = 0.4$$

Question 6

A deck of cards contains 52 cards. What is the probability of drawing either a heart or a king from the deck? (There are 13 hearts and 4 kings, with 1 card being both a heart and a king.)

Accepted Answer

$$P(A 	ext{ or } B) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}$$

Question 7

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

Accepted Answer

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

Question 8

In a lottery, the probability of winning is 0.01. What are the actual odds against winning?

Accepted Answer

$$	ext{Odds against} = \frac{P(	ext{not win})}{P(	ext{win})} = \frac{0.99}{0.01} = 99:1$$

Question 9

A bag contains 5 red balls and 3 blue balls. If two balls are drawn without replacement, what is the probability that both are red?

Accepted Answer

$$P(	ext{both red}) = \frac{5}{8} 	imes \frac{4}{7} = \frac{20}{56} = \frac{5}{14}$$

Question 10

A password consists of 3 letters, each chosen from the 26 English letters. How many different passwords are possible if letters can be repeated?

Accepted Answer

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

Question 11

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

Accepted Answer

$$_{10}C_4$ = \frac{10!}{(10-4)!4!} = \frac{10!}{6!4!} = 210$$

Question 12

A word is formed from the letters in 'SUCCESS'. How many different arrangements are possible?

Accepted Answer

$$\frac{7!}{3!2!2!} = 210$$

Question 13

A student wants to know the probability of getting at least one question correct on a 5-question multiple-choice quiz (each question has 4 choices, only one correct, and guesses randomly). What is the probability?

Accepted Answer

$$P(	ext{at least one correct}) = 1 - (\frac{3}{4})^5$ \approx 1 - 0.237 = 0.763$$

Question 14

If the probability of event A is 0.3 and the probability of event B is 0.4, and the probability that both occur is 0.1, what is the probability that either A or B occurs?

Accepted Answer

$$P(A 	ext{ or } B) = 0.3 + 0.4 - 0.1 = 0.6$$

Question 15

A researcher wants to estimate the probability of a rare disease using a simulation. Which approach to probability is being used?

Accepted Answer

Simulation approach

Question 16

A gambler believes that after losing five times in a row, the probability of winning the next game is higher. Which statistical principle does this violate?

Accepted Answer

Law of large numbers applies to many trials, not individual outcomes.

Question 17

If two events A and B are disjoint (mutually exclusive), which of the following is true?

Accepted Answer

$$P(A 	ext{ and } B) = 0$$

Question 18

A prior probability is:

Accepted Answer

An initial probability value obtained before any additional information is available.

Question 19

A box contains 10 bulbs, 2 of which are defective. If two bulbs are drawn at random without replacement, what is the probability that both are defective?

Accepted Answer

$$P(	ext{both defective}) = \frac{2}{10} 	imes \frac{1}{9} = \frac{2}{90} = \frac{1}{45}$$

Probability and Counting Rules in Introductory Statistics

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