Q: A car's security system access code consists of four digits, each digit can be any number from 0 through 9, and digits can be repeated. How many possible access codes are there?

$$10 \times 10 \times 10 \times 10 = 10,000$$

Q: A building contractor is planning to develop a subdivision with 6 one-story houses, 4 two-story houses, and 2 split-level houses. In how many distinguishable ways can the houses be arranged?

$$\frac{12!}{6! \times 4! \times 2!} = 13,860$$

Q: What is the probability that a randomly selected senior was matched with a residency position and it was one of the senior’s top three choices, given $$P(A) = \frac{18,108}{19,326}$$ and $$P(B|A) = 0.756$$?

$$P(A \text{ and } B) = P(A) \cdot P(B|A) = \frac{18,108}{19,326} \times 0.756 = 0.708$$

Q: Which formula represents the number of combinations of n objects taken r at a time?

$$nCr = \frac{n!}{(n-r)! \times r!}$$

Q: A lottery requires you to choose 5 numbers from 1 to 50, and the order does not matter. How many possible combinations are there?

$$\frac{50!}{(50-5)! \times 5!}$$

Q: A password consists of 3 letters followed by 2 digits. If letters and digits can be repeated, how many possible passwords are there? (Assume 26 letters and 10 digits)

$$26 \times 26 \times 26 \times 10 \times 10 = 1,757,600$$

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

$$\frac{10!}{(10-4)! \times 4!}$$

Q: A four-digit code is formed from the digits 0-9, with no digit repeated and the first digit cannot be 0 or 1. How many possible codes are there?

$$8 \times 10 \times 10 \times 10 = 8,000$$

Question 1

A probability experiment involves flipping a coin and rolling a six-sided die. How many possible outcomes are there in the sample space for this experiment?

Accepted Answer

$$12$$

Question 2

A car's security system access code consists of four digits, each digit can be any number from 0 through 9, and digits can be repeated. How many possible access codes are there?

Accepted Answer

$$10 	imes 10 	imes 10 	imes 10 = 10,000$$

Question 3

If you roll a six-sided die, what is the probability of rolling a number less than 5?

Accepted Answer

$$\frac{4}{6} = 0.667$$

Question 4

A deck of 52 cards is used. What is the probability that the second card drawn is a queen, given that the first card drawn is a king and is not replaced?

Accepted Answer

$$\frac{4}{51} = 0.078$$

Question 5

Which of the following best describes mutually exclusive events?

Accepted Answer

Events that cannot occur at the same time and have no outcomes in common

Question 6

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

$$0.9$$

Question 7

A building contractor is planning to develop a subdivision with 6 one-story houses, 4 two-story houses, and 2 split-level houses. In how many distinguishable ways can the houses be arranged?

Accepted Answer

$$\frac{12!}{6! 	imes 4! 	imes 2!} = 13,860$$

Question 8

What is the probability that a randomly selected senior was matched with a residency position and it was one of the senior’s top three choices, given $$P(A) = \frac{18,108}{19,326}$$ and $$P(B|A) = 0.756$$?

Accepted Answer

$$P(A 	ext{ and } B) = P(A) \cdot P(B|A) = \frac{18,108}{19,326} 	imes 0.756 = 0.708$$

Question 9

If the probability of event E is $$0.25$$, what is the probability of the complement of E?

Accepted Answer

$$0.75$$

Question 10

Which formula represents the number of combinations of n objects taken r at a time?

Accepted Answer

$$nCr = \frac{n!}{(n-r)! 	imes r!}$$

Question 11

A lottery requires you to choose 5 numbers from 1 to 50, and the order does not matter. How many possible combinations are there?

Accepted Answer

$$\frac{50!}{(50-5)! 	imes 5!}$$

Question 12

If the probability of event A is $$0.6$$ and the probability of event B is $$0.5$$, and A and B are independent, what is the probability that both A and B occur?

Accepted Answer

$$0.3$$

Question 13

Which of the following is NOT a type of probability described in the study materials?

Accepted Answer

Absolute probability

Question 14

A password consists of 3 letters followed by 2 digits. If letters and digits can be repeated, how many possible passwords are there? (Assume 26 letters and 10 digits)

Accepted Answer

$$26 	imes 26 	imes 26 	imes 10 	imes 10 = 1,757,600$$

Question 15

If the frequency of event E is 15 and the total frequency is 60, what is the empirical probability of E?

Accepted Answer

$$\frac{15}{60} = 0.25$$

Question 16

Which equation correctly expresses the relationship between the probability of an event and its complement?

Accepted Answer

$$P(E) + P(E') = 1$$

Question 17

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

Accepted Answer

$$\frac{10!}{(10-4)! 	imes 4!}$$

Question 18

If event A is 'rolling a 3 on a die' and event B is 'rolling a 4 on a die', what is $$P(A 	ext{ and } B)$$?

Accepted Answer

$$0$$

Question 19

Which of the following best describes the fundamental counting principle?

Accepted Answer

If one event can occur in m ways and a second event can occur in n ways, the number of ways the two events can occur in sequence is $$m 	imes n$$

Question 20

A four-digit code is formed from the digits 0-9, with no digit repeated and the first digit cannot be 0 or 1. How many possible codes are there?

Accepted Answer

$$8 	imes 10 	imes 10 	imes 10 = 8,000$$

Probability and Counting Rules: Fundamental Concepts and Applications

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