Q: A card is drawn from a standard deck. What is the probability that it is a heart or a king?

$$P(\text{heart or king}) = P(\text{heart}) + P(\text{king}) - P(\text{heart and king}) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}$$

Q: If two light bulbs are selected at random from a box of 100 bulbs (without replacement), and 5 are defective, what is the probability that both bulbs are defective? (Use the multiplication rule.)

$$P(\text{both defective}) = \frac{5}{100} \times \frac{4}{99} = \frac{20}{9900} = \frac{2}{990}$$

Q: A password consists of 3 letters followed by 2 digits. How many different passwords are possible if letters and digits can be repeated?

$$26^3$$ \times $$10^2 = 17$$,576 \times 100 = 1,757,600$$

Q: A coin is flipped 5 times. What is the probability of getting at least one head?

$$P(\text{at least one head}) = 1 - P(\text{no heads}) = 1 - (\frac{1}{2})^5$ = 1 - \frac{1}{32} = \frac{31}{32}$$

Question 1

A pharmaceutical company tests a new drug on 200 patients and finds that 30 experience a certain side effect. Using the relative frequency approach, what is the probability that a randomly selected patient will experience the side effect?

Accepted Answer

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

Question 2

A fair die is rolled. What is the probability of rolling a number greater than 4 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 of an event approaches the actual probability.

Question 4

A student claims that the probability of passing the next statistics test is $$0.5$$ because they will either pass or fail. What is wrong with this reasoning?

Accepted Answer

The outcomes are not necessarily equally likely; the actual probability depends on factors like preparation and test difficulty.

Question 5

If the probability of event A is $$0.3$$, what is the probability of the complement of A?

Accepted Answer

$$P(\overline{A}) = 1 - 0.3 = 0.7$$

Question 6

A medical test for a rare disease has a $$1\%$$ false positive rate. If 1000 healthy people are tested, what is the expected number of false positives?

Accepted Answer

$$1000 	imes 0.01 = 10$$

Question 7

A card is drawn from a standard deck. What is the probability that it is a heart or a king?

Accepted Answer

$$P(	ext{heart or king}) = P(	ext{heart}) + P(	ext{king}) - P(	ext{heart and king}) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}$$

Question 8

If two light bulbs are selected at random from a box of 100 bulbs (without replacement), and 5 are defective, what is the probability that both bulbs are defective? (Use the multiplication rule.)

Accepted Answer

$$P(	ext{both defective}) = \frac{5}{100} 	imes \frac{4}{99} = \frac{20}{9900} = \frac{2}{990}$$

Question 9

A password consists of 3 letters followed by 2 digits. How many different passwords are possible if letters and digits can be repeated?

Accepted Answer

$$26^3$$ 	imes $$10^2 = 17$$,576 	imes 100 = 1,757,600$$

Question 10

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

Accepted Answer

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

Question 11

A coin is flipped 5 times. What is the probability of getting at least one head?

Accepted Answer

$$P(	ext{at least one head}) = 1 - P(	ext{no heads}) = 1 - (\frac{1}{2})^5$ = 1 - \frac{1}{32} = \frac{31}{32}$$

Question 12

Which of the following statements about independent events is correct?

Accepted Answer

If events A and B are independent, then $$P(A 	ext{ and } B) = P(A) 	imes P(B).$$

Question 13

A bag contains 3 red, 2 blue, and 5 green marbles. If two marbles are drawn at random without replacement, what is the probability that both are blue?

Accepted Answer

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

Question 14

A password consists of 4 letters, where two are A's and two are B's. How many different arrangements are possible?

Accepted Answer

$$\frac{4!}{2!2!} = 6$$

Question 15

A hospital administrator wants to know the probability that, in a group of 5 patients, none will develop a certain infection if the probability for each patient is $$0.1$$ and patients are independent. What is the probability that at least one patient develops the infection?

Accepted Answer

$$P(	ext{at least one}) = 1 - (0.9)^5$ = 1 - 0.59049 = 0.40951$$

Question 16

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

Accepted Answer

$$P(A 	ext{ or } B) = P(A) + P(B) = 0.4 + 0.5 = 0.9$$

Question 17

A box contains 8 light bulbs, 3 of which are defective. If two bulbs are selected at random without replacement, what is the probability that at least one is defective?

Accepted Answer

$$P(	ext{at least one defective}) = 1 - P(	ext{none defective}) = 1 - \frac{5}{8} 	imes \frac{4}{7} = 1 - \frac{20}{56} = 1 - \frac{5}{14} = \frac{9}{14}$$

Question 18

A lottery requires you to pick 6 numbers from 49. How many different possible combinations of numbers are there?

Accepted Answer

$$_{49}C_6$ = \frac{49!}{(49-6)!6!} = 13,983,816$$

Question 19

Which of the following is an example of a compound event?

Accepted Answer

Drawing a red card and a king from a deck of cards.

Probability Rules and Counting Principles in Introductory Statistics

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