Q: How many different four-digit numbers can be formed from the digits 0, 1, 2, 5, 6, and 9 if each digit can be used only once and the first digit cannot be zero?

$5 \times 5 \times 4 \times 3 = 300$

Q: If two events $A$ and $B$ are independent, which of the following must be true?

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

Q: Suppose three machines, $B_1$, $B_2$, and $B_3$, make 30%, 45%, and 25% of the products, respectively. If 2%, 3%, and 2% of the products made by each machine are defective, what is the probability that a randomly selected product is defective?

$0.3 \times 0.02 + 0.45 \times 0.03 + 0.25 \times 0.02 = 0.0245$

Q: A hand consists of 5 cards. What is the probability of holding 3 jacks and 2 queens?

$\frac{\binom{4}{3} \times \binom{4}{2}}{\binom{52}{5}}$

Question 1

Which of the following best describes a sample space in probability theory?

Accepted Answer

The set of all possible outcomes of a statistical experiment.

Question 2

Given the sample space $S = \{1,2,3,4,5,6\}$ for rolling a die, which of the following is the correct representation of the event 'prime number'?

Accepted Answer

$\{2,3,5\}$

Question 3

If $A = \{1,4,5,6,7\}$ and $B = \{2,4,8\}$, what is $A \cap B$?

Accepted Answer

$\{4\}$

Question 4

How many different four-digit numbers can be formed from the digits 0, 1, 2, 5, 6, and 9 if each digit can be used only once and the first digit cannot be zero?

Accepted Answer

$5 	imes 5 	imes 4 	imes 3 = 300$

Question 5

A president and a treasurer are to be chosen from a student club consisting of 50 people. How many different choices of officers are possible if there are no restrictions?

Accepted Answer

$50 	imes 49 = 2450$

Question 6

What is the number of ways to arrange 5 different trees in a row?

Accepted Answer

$5! = 120$

Question 7

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

Accepted Answer

$P(A \cap B) = 0$

Question 8

A coin is tossed twice. What is the probability that at least one head occurs?

Accepted Answer

$\frac{3}{4}$

Question 9

Given $P(A) = 0.3$, $P(B) = 0.5$, and $P(A \cap B) = 0.1$, what is $P(A \cup B)$?

Accepted Answer

$0.3 + 0.5 - 0.1 = 0.7$

Question 10

Suppose the probability that a randomly selected cable is longer than 2000 millimeters is $0.92$. What is the probability that a randomly selected cable is not longer than 2000 millimeters?

Accepted Answer

$1 - 0.92 = 0.08$

Question 11

If two events $A$ and $B$ are independent, which of the following must be true?

Accepted Answer

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

Question 12

A box contains 4 white balls and 3 black balls. A second box contains 3 white balls and 5 black balls. One ball is drawn from the first box and placed unseen into the second box. Then a ball is drawn from the second box. What is the probability that the ball now drawn from the second box is black?

Accepted Answer

$\frac{1}{2}$

Question 13

In a college football training session, the defensive coordinator needs to have 10 players standing in a row. Among these players, there are 1 freshman, 2 sophomores, 4 juniors, and 3 seniors. In how many ways can they be arranged in a row if only their class level will be distinguished?

Accepted Answer

$\frac{10!}{1!2!4!3!}$

Question 14

Suppose three machines, $B_1$, $B_2$, and $B_3$, make 30%, 45%, and 25% of the products, respectively. If 2%, 3%, and 2% of the products made by each machine are defective, what is the probability that a randomly selected product is defective?

Accepted Answer

$0.3 	imes 0.02 + 0.45 	imes 0.03 + 0.25 	imes 0.02 = 0.0245$

Question 15

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

Accepted Answer

$0.4 	imes 0.5 = 0.2$

Question 16

A hand consists of 5 cards. What is the probability of holding 3 jacks and 2 queens?

Accepted Answer

$\frac{\binom{4}{3} 	imes \binom{4}{2}}{\binom{52}{5}}$

Question 17

Suppose the probability that a regularly scheduled flight departs on time is $P(D) = 0.83$, the probability that it arrives on time is $P(A) = 0.82$, and the probability that it departs and arrives on time is $P(D \cap A) = 0.78$. What is the probability that a plane arrives on time given that it departed on time?

Accepted Answer

$\frac{P(D \cap A)}{P(D)} = \frac{0.78}{0.83} \approx 0.94$

Question 18

If $A$ and $B$ are complementary events, which equation is correct?

Accepted Answer

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

Question 19

Suppose the probabilities that a person will purchase a new automobile and choose the color green, white, red, or blue are 0.09, 0.15, 0.21, and 0.23, respectively. What is the probability that a given buyer will purchase a new automobile that comes in one of these four colors?

Accepted Answer

$0.09 + 0.15 + 0.21 + 0.23 = 0.68$

Introduction to Probability: Structured Study Notes

Study Guide - Practice Questions

Study Guide - Flashcards