Quantitative Reasoning Study Guide for the CLT

General Information

The questions in the Classic Learning Test (CLT) Quantitative Reasoning section assess your math abilities in these three areas:

algebra I and algebra II
geometry and trigonometry
mathematical reasoning involving algebra and geometry

There are 40 questions in this section and you will have 45 minutes to complete them. You will not be allowed to use a calculator on any part of this test, but you will have access to the following formula chart for every question:

Area of a circle = $\pi r^2$, where $r$ is the radius of the circle

Circumference of a circle = $2\pi r$, where $r$ is the radius of the circle

There are $360$ degrees in a circle.

There are $2\pi$ radians in a circle.

Volume of a sphere = $\frac{4}{3}\pi r^3$, where $r$ is the radius of the sphere

Surface area of a sphere = $4\pi r^2$, where $r$ is the radius of the sphere

Area of a rectangle = length $\times$ width

Area of a triangle = $\frac{1}{2}$ (base $\times$ height)

The sum of the measures of the interior angles of a triangle is $180^{\circ}$.

Pythagorean theorem (for a right triangle): If $a$, $b$, and $c$ are the side lengths of the triangle, and $c$ is the hypotenuse, then $a^2 + b^2 = c^2$ .

$30^{\circ}–60^{\circ}–90^{\circ}$ triangles have side lengths in a ratio of $1 \text{:} \sqrt{3} \text{:} 2$, corresponding to their opposite angle.

$45^{\circ}–45^{\circ}–90^{\circ}$ triangles have side lengths in a ratio of $1 \text{:} 1 \text{:} \sqrt{2}$, corresponding to their opposite angle.

In this study guide, you will review the most important elements tested in the CLT Quantitative Reasoning section.

Algebra

Algebra is a branch of mathematics that uses symbols, which represent numbers and values in formulas and equations. These symbols allow for the expression of abstract mathematical relationships.

Expressions

An expression in algebra is a combination of variables, numbers, and operations that represent a specific value or relationship. Expressions are not equations, so they do not have an equality sign, and they cannot be “solved.” Instead, they can be simplified or evaluated when the values of variables are known.

These are some examples of algebraic expressions:

\[3x + 5\] \[2y^2 - 4y + 7\] \[a + b - 9\] \[5m \times 4n\]

Simplifying Expressions

To simplify an expression is to reduce it to its most concise form.

Properties of Exponents

One way to simplify an expression is using properties of exponents. These are the most common properties of exponents:

exponent product rule—The exponent product rule is a way to simplify the product of two exponents with the same base. The rule is as follows:

\[x^a \times x^b = x^{a + b}\]

power of a product rule—The power of a product rule allows us to break apart the product of two terms raised to a power into two separate terms raised to a power and then multiplied together. The rule is:

\[(xy)^a = x^a \times y^a\]

power of a quotient rule—The power of a quotient rule allows us to break apart a fraction raised to a power into a fraction where the numerator and denominator are separately raised to a power. The rule is:

\[\left( \frac{x}{y} \right)^a = \frac{x^a}{y^a}\]

exponent quotient rule—The exponent quotient rule is a way to simplify the quotient of two exponents with the same base. The rule is as follows:

\[\frac{x^a}{x^b} = x^{a - b}\]

negative exponent rule—The negative exponent rule is a way to turn a negative exponent into a positive one. The rule is:

\[x^{-a} = \frac{1}{x^a}\]

power of a power exponent rule—The power of a power exponent rule is a way to simplify an expression involving two powers. The rule is as follows:

\[(x^a)^b = x^{a \times b}\]

zero exponent rule—The zero exponent rule simply tells us that any number not equal to zero that is raised to the zero power is equal to $1$. For example:

\[5^0 = 1, 4.5^0 = 1\]

radical to exponent rule—The radical to exponent rule is a way to turn a radical into an exponent, which can be helpful if we want to apply some of the other properties of exponents listed above. The rule is:

\[\sqrt[b]{x^a} = x^{\frac{a}{b}}\]

Note: You can also go the other way and transform an exponent to a radical.

Rationalizing the Denominator

Another simplification strategy is known as rationalizing the denominator. When there is a square root in the denominator of an expression, we multiply the numerator and denominator by this square root to clear the denominator of any square roots.

For example, suppose we want to rationalize the denominator of $\frac{1}{\sqrt{2}}$. Then, multiplying the top and bottom by $\sqrt{2}$, we have:

$\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{\sqrt{2}}{2}$.

In the last step, we use exponent properties to simplify the expression further.

Other Methods for Simplifying

Often, to simplify expressions, we will need to find a common denominator for two numbers, $a$ and $b$. In order to do this, we need to find the least common multiple (LCM) of $a$ and $b$. It can be useful to find the prime factorization of each number first. To do this, we find a prime number that divides $a$, then we divide $a$ by that number and repeat until $a$ is itself a prime number. For example, suppose $a = 12$:

Since $2$ divides $12$, we can perform this division to get $12 \div 2 = 6$. Since $2$ divides $6$, we can perform this division to get $6 \div 2 = 3$.

Now, we are left with a prime number, $3$, so we stop. Therefore, we can write $12$ as a product of prime numbers: $12 = 2 \times 2 \times 3 = 2^2 \times 3$. This is the prime factorization for $a$.

Now, suppose we have the prime factorization for both $a$ and $b$. Then, the LCM of $a$ and $b$ can be found by taking the product of the highest power of each prime of $a$ and $b$. For example, if $a = 12$ and $b = 9$, then the prime factorization of these numbers is:

\[12 = 2^2 \times 3 = 2^2 \times 3^1\] \[9 = 3^2\]

The highest power of $2$ between the numbers is $2^2$ and the highest power of $3$ between the numbers is $3^2$. So, we take the product of $2^2$ and $3^2$ to get that the LCM of $12$ and $9$ is $2^2 \times 3^2 = 4 \times 9 = 36$.

Other ways to simplify expressions include combining like terms and applying arithmetic operations.

For example, to simplify the expression $3x + 2x$, we add the coefficients and include the $x$ with the sum:

\[3x + 2x = (3+2)x = 5x\]

Think of it as adding three apples and two apples to get five apples.

To simplify the algebraic expression $2x^2 \times x^3$, apply the exponent product rule $(x^a \times x^b = x^{a+b})$ to get:

\[2x^2 \times x^{3}\] \[= 2 x^{2+3}\] \[= 2x^{5}\]

In a more complex case, to simplify $\frac{4x^3 \times 2x^2}{x^{4}}$, first, multiply the coefficients ($4$ and $2$) in the numerator to get $4 \times 2 = 8$ and add the exponents, using the rule above:

\[\frac{4x^3 \times 2x^2}{x^{4}}\] \[= \frac{(4 \times 2)x^{3+2}}{x^{4}}\] \[= \frac{8 x^{5}}{x^{4}}\]

Now, we use the exponent quotient rule ($x^a \div x^{b} = x^{a-b}$) to simplify the rest of it:

\[= 8 x^{5-4}\] \[= 8x^{1}\] \[= 8x\]

Substituting Terms or Values and Simplifying

To substitute is to replace variables in an expression with given values to evaluate the expression. Once substituted, the expression can be simplified through basic arithmetic.

For example, if you are given the expression $3x + 5(x-2)$ and told $x = 2$, you will substitute $2$ into the expression to get:

\[3(2)+5(2-2)\] \[= 6 + 5(0)\] \[= 6 + 0\] \[= 6\]

For the more complex expression $2a^2 - 4b + c$, where $a = 3$, $b = 1$, and $c = 5$, substitute the values to get:

\[2(3)^{2} - 4(1) + (5)\] \[= 2(9) - 4 + 5\] \[= 18 - 4 + 5\] \[= 19\]

In cases where the algebraic expression is long and complicated, usually you will be able to simplify the expression first and then substitute the values to get your answer.

Equations

Equations are mathematical statements that assert the equality of two expressions. In essence, an equation contains an equal sign ($=$), showing that what is on either side is of equal value. An equation involves variables, constants, and operators, which equates two expressions. Solving equations means finding the value of the variables that satisfy the equation.

There are many types of equations, such as linear, quadratic, cubic, exponential, rational, etc. Let’s look at some of them:

A linear equation is a simple equation where the variable is raised to the power of $1$, such as $2x + 3 = 7$. Solving it, we get:

\[2x + 3 = 7\] \[2x = 7 - 3\] \[2x = 4\] \[x = \frac{4}{2} = 2\]

Note: An equation for a straight line, such as $y = 3x + 1$, is a linear equation.

A rational equation is an equation that contains one or more rational expressions, such as $\frac{2}{3x} = 4$.
An exponential equation is an equation that has the variable within the exponent, such as $3^x = 27$.

Substituting Values Using Special Symbols

Special symbols are often used to represent operations or functions in equations, and substituting values into these can be a key skill. For example, problems might involve symbols like $\oplus$ or $\star$ to represent custom operations.

To solve these types of problems, you must:

Understand what the special symbols represent.
Know how to substitute given values.
Be comfortable performing basic arithmetic operations.

Let’s look at an example:

If $a \oplus b = a^2 - b$, what is the value of $3 \oplus 2$?

Solution

Putting $3$ into $a$ and $2$ into $b$, we find the answer as follows:

\[3 \oplus 2 = (3)^{2} - (2) = 9 - 2 = 7\]

Here, the special symbol $\oplus$ indicates squaring the first number and subtracting the second. Understanding this is key to correctly solving these types of problems.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, meaning they include a term with $x^2$. The standard form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. The equation is solved by finding the values of $x$ that make the equation true (there can be up to two values). These are examples of quadratic equations:

\[x^2 + 4x + 10 = 0\] \[2x^2 - 8 = 10\]

Using factoring or the quadratic formula, we can solve a quadratic. The solution to a quadratic equation can include real or complex roots. If the solution to a quadratic includes a term with the square root of a negative number, this solution is said to be a complex root. Otherwise, the solution is a real root. When dealing with the square root of a negative number, say $\sqrt{-a}$, we can use the properties of exponents above to rewrite this as $\sqrt{-1}\sqrt{a}$. Since $\sqrt{-1}$ comes up quite often, mathematicians have given this a name: $\sqrt{-1} = i$. With this in mind, we can write $\sqrt{-a}$ as $i\sqrt{a}$. Any number involving $i$ is called an imaginary number.

Factoring and Solving

Factoring is a common method for solving quadratic equations by expressing the quadratic as a product of two binomials.

To solve a quadratic equation using the factoring method:

Write the equation in the form $ax^2 + bx + c = 0$.
Look for two numbers that multiply to $ac$ and add to $b$.
Rewrite the middle term using these numbers and factor by grouping.

An example will make this process clearer.

Solve:

\[x^2 + 5x + 6 = 0\]

Solution

First, find two numbers that multiply to $6$ and add to $5$. Those two numbers are $2$ and $3$. Now, we can rewrite the middle term using these numbers:

\[x^2 + 2x + 3x + 6 = 0\]

Next, we can factor an $x$ from the first two terms and a $3$ from the last two terms:

\[x(x + 2) + 3(x + 2) = 0\]

Finally, we can factor as follows:

\[(x + 2)(x + 3) = 0\]

Solving for $x$, we know $x+2 = 0$ and $x + 3 = 0$. Thus, there are two solutions, with $x = -2$ and $x = -3$.

Quadratic Equations with Real and Complex Roots

We can also solve a quadratic equation in the form of $ax^2 + bx + c = 0$ using the quadratic formula:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

The part underneath the square root, $b^2 - 4ac$, is known as the discriminant. The discriminant determines the nature of the roots:

If $b^2 - 4ac > 0$, the equation has two distinct real roots.
If $b^2 - 4ac = 0$, the equation has one real root (a repeated root).
If $b^2 - 4ac < 0$, the equation has two complex roots.

Let’s try an example.

Find the nature of the roots of the quadratic equation $x^2 + 4x + 5 = 0$ and solve it.

We will find the value of the discriminant first:

\[b^2 - 4ac\] \[= (4)^{2} - 4(1)(5)\] \[= 16 - 20\] \[= -4\]

So, even before solving the equation, we know that it will have complex roots. Now, let’s solve this using the quadratic formula:

\[x = \frac{-4 \pm \sqrt{4^2 - 4(1)(5)}}{2(1)} = \frac{-4 \pm \sqrt{16 - 20}}{2}\] \[x = \frac{-4 \pm \sqrt{-4}}{2} = \frac{-4 \pm 2i}{2}\]

Thus, the solutions are $x = -2 + i$ and $x = -2 - i$, which are complex roots.

Systems of Equations

A system of equations consists of two or more equations with the same set of variables. The goal is to find a common solution for all equations.

This is an example of a two-variable system:

$\left\{ \begin{array}{c} 2x + 3y = 6 \\ x - y = 1 \end{array} \right.$$x - y = 1$

This is an example of a three-variable system:

\[\left\{ \begin{array}{c} x + y + z = 6 \\ 2x - y + z = 4 \\ 3x + 2y - z = 5 \end{array} \right.\]

We can use the following methods to solve a system of equations:

substitution—With this method, you solve one equation for one variable, then substitute it into the other equation(s).
elimination—Here, you add or subtract equations to eliminate one variable, then solve for the remaining variables.
graphing—You simply plot each equation on a graph and find the point(s) where they intersect.

Let’s try examples of each method.

Substitution Method

Solve:

\[\left\{ \begin{array}{c} 2x + y = 7 \\ x - y = 3 \end{array} \right.\]

Solution

First, solve the second equation for $x$:

\[x - y = 3\] \[x = y + 3\]

Now, substitute this into the first equation:

\[2x + y = 7\] \[2(y + 3) + y = 7\]

Simplify and solve for $y$:

\[2y + 6 + y = 7\] \[3y = 1\] \[y = \frac{1}{3}\]

Lastly, substitute $y = \frac{1}{3}$ back into one of the original equations to find $x$:

\[x - y = 3\] \[x = 3 + \frac{1}{3}\] \[x = \frac{10}{3}\]

Elimination Method

For the set of equations above, we can use the elimination method as well:

\[\left\{ \begin{array}{c} 2x + y = 7 \\ x - y = 3 \end{array} \right.\]

Adding both equations, we have:

\[3x = 10\]

Note that the $y$s cancel out. Solving for $x$, we get:

\[x = \frac{10}{3}\]

To find $y$, we substitute the value of $x$ into one of the original equations from above:

\[x - y = 3\] \[\frac{10}{3} - y = 3\] \[y = \frac{10}{3} -3 = \frac{1}{3}\]

Graphing Method

For the system of equations above, we can also solve it using the graphing method. We will graph both the equations (lines), and their intersecting point is the solution of the system. The graphs are shown below:

1 Graphing to Solve.png

Inequalities

An inequality is a statement that compares two expressions by using inequality symbols. Unlike equations, where two expressions are exactly equal, inequalities indicate a range of possible values rather than one fixed value. There are five inequality symbols:

greater than ($\boldsymbol{>}$)—This symbol indicates that the value on the left side of the inequality is larger than the value on the right. For example, $x > 5$ means $x$ is any number greater than $5$.
less than ($\boldsymbol{<}$)—This symbol shows that the value on the left is smaller than the one on the right. For example, $y < 10$ means $y$ is any number smaller than $10$.
greater than or equal to ($\boldsymbol{\geq}$)—This indicates the value on the left is either greater than or equal to the value on the right. For example, $z \geq -3$ means $z$ can be any number greater than or equal to $-3$.
less than or equal to ($\boldsymbol{\leq}$)—This symbol indicates that the left-side value is either smaller than or equal to the right-side value. For example, $a \leq 7$ means $a$ is any number less than or equal to $7$.
Not equal to ($\boldsymbol{\neq}$)—This symbol tells us that the value on the left is not equal to the value on the right. For example, $b \neq 10$ means that $b$ can be any number other than $10$.

Here are some examples of inequalities:

\[x + 3 > 7\] \[5y \leq 20\] \[-2z \geq 8\] \[x - 7 < 12\] \[14 \neq 41\]

Solving Inequalities

When solving inequalities, the process is similar to solving equations, but with one key difference: If you multiply or divide both sides of the inequality by a negative number, the inequality symbol must be flipped.

Here are the steps to solve an inequality:

Isolate the variable on one side of the inequality.
Perform arithmetic operations (addition, subtraction, multiplication, or division) on both sides just as you would in an equation.
If you multiply or divide by a negative number, reverse the inequality sign.
Express the solution in the form of a range or on a number line.

Let’s do an example.

Solve:

\[2x - 5 > 3\]

Solution

First, add $5$ to both sides:

\[2x - 5 + 5 > 3 + 5\] \[2x > 8\]

Next, divide both sides by $2$:

\[\frac{2x}{2} > \frac{8}{2}\] \[x > 4\]

The solution is $x > 4$, meaning any value of $x$ greater than $4$ satisfies the inequality.

Systems of Inequalities

A system of inequalities consists of two or more inequalities that are solved together. Solutions to a system of inequalities are the values that satisfy all inequalities in the system simultaneously.

Below, we see a system of two inequalities:

\[\left\{ \begin{array}{c} y \leq 2x + 3 \\ y > -x - 1 \end{array} \right.\]

To solve the system, first, we will graph the line $y = 2x + 3$ and shade the region below the line. The $\leq$ symbol corresponds to the region that is on the line and below.

Next, we will graph the line $y = -x -1$ in the same coordinate plane and shade the region that is above the line. The inequality symbol $>$ corresponds to the region that is above the line.

The solution is the overlapping region where both inequalities are true:

2 Inequality System Graph A.png

With a system of three inequalities, the process is exactly the same, but we have to deal with three inequalities rather than two. A system of three inequalities is shown below:

\[\left\{ \begin{array}{c} x - y < 7 \\ y \leq 4 - 2x \\ x \geq - 4 \end{array} \right.\]

The solution set of the system is shown below:

3 Inequality Graph B.png

Compound Inequalities

A compound inequality consists of two separate inequalities joined by the words and or or. These inequalities express a range of values that satisfy one or both conditions. For instance, these are compound inequalities:

\[x > 2 \text{ and } x \leq 7\] \[2x > 4 \text{ or } 2x \geq 2\]

Note: The “and” may sometimes be left out. For instance, the above compound inequality could be written as $2 < x \leq 7$

Here are guidelines for solving a compound inequality:

Solve each inequality separately.
If the compound inequality uses “and,” find the intersection (overlap) of the solutions.
If the compound inequality uses “or,” find the union of the solutions.

Solving an example will make the process clearer.

Solve:

\[1 < 2x + 3 \leq 7\]

Solution

First, break apart the compound inequality into two separate inequalities:

\[1 < 2x + 3 \text{ and } 2x + 3 \leq 7\]

Solving the first one, we get:

\[1 < 2x + 3\] \[-2 < 2x\] \[-1 < x\]

Solving the second one, we get:

\[2x + 3 \leq 7\] \[2x \leq 4\] \[x \leq 2\]

Combining both solutions, we have $-1 < x \leq 2$, meaning $x$ is any number between $-1$ and $2$, not including $-1$ but including $2$.

Absolute Value

Absolute value refers to the distance of a number from zero on the number line, always expressed as a non-negative number. It is denoted by two vertical bars, $\vert x \vert$, and represents the magnitude of $x$ without regard to its sign. Whether a number is positive or negative, its absolute value measures how far it is from zero. This concept is widely used in solving equations and inequalities involving distances.

Simplifying Absolute Value Expressions

Simplifying absolute value expressions involves reducing the values within the bars and then applying the absolute value. The steps are as follows:

Simplify the expression inside the absolute value bars.
Apply the absolute value to convert any negative result to positive.
Perform any additional operations outside the bars.

Let’s look at an example.

Simplify:

\[\vert 3 - 7 \vert + 5\]

Solution

Simplify the expression inside the bars:

\[\vert 3 - 7 \vert = \vert -4 \vert\]

Apply the absolute value so that the number becomes positive:

\[\vert -4 \vert = 4\]

Now, add all the values to simplify:

\[4 + 5 = 9\]

Solving Absolute Value Equations

Solving absolute value equations requires considering both the positive and negative cases of the expression inside the absolute value. Follow these guidelines:

Set up two equations, one where the expression inside the bars is equal to the positive value and one where it is equal to the negative value.
Solve each equation separately.
Check the solutions in the original equation.

Let’s solve an equation.

Solve:

\[|x - 2| = 5\]

Solution

If we dissect the absolute value equation, we have the following two equations:

\[x - 2 = 5 \text{ and } x - 2 = -5\]

We will solve the first equation for $x$:

\[x - 2 = 5\] \[x = 7\]

Then, we will solve the second one:

\[x - 2 = -5\] \[x = -3\]

Last, we will check these:

\[|7 - 2| \stackrel{?}{=} 5\] \[5 = 5\]

And

\[|-3 - 2| \stackrel{?}{=} 5\] \[| -5 | \stackrel{?}{=} 5\] \[5 = 5\]

So, the solutions are $x = 7$ and $x = -3$.

Solving Absolute Value Inequalities

Absolute value inequalities are solved by breaking the inequality into two separate inequalities. The steps differ slightly depending on whether the inequality involves less than or greater than.

For inequalities like $$

< a $$:

Set up a compound inequality: $-a < x < a$.
Solve the inequality for $x$.

For inequalities like $$

> a $$:

Set up two separate inequalities: $x > a \,\text{or} \,x < -a$.
Solve each inequality for $x$.

Let’s do an example.

Solve:

\[|x - 3| < 4\]

Solution

First, we set up the compound inequality:

\[-4 < x - 3 < 4\]

Add $3$ to all three parts to get the solution of the inequality:

\[-4 + 3 < x - 3 + 3 < 4 + 3\] \[-1 < x < 7\]

Patterns and Sequences

Patterns and sequences are foundational concepts that help us predict or understand the arrangement of numbers, shapes, or objects based on a set of rules. A pattern is a recognizable, repeated structure, while a sequence is a list of numbers or objects arranged in a specific order, often following a pattern. Both patterns and sequences allow us to analyze and find relationships in sets of data, often helping us solve problems by predicting what comes next.

Identifying Patterns

There are several common types of patterns that we frequently encounter in sequences:

arithmetic sequences—In arithmetic sequences, the difference between consecutive terms is constant. For example, in the sequence $2, \, 4, \, 6, \, 8...$ the difference between each number is $2$. This is a pattern in which the common difference (the difference between a term and its previous term) is $2$.
geometric sequences—In geometric sequences, each term is obtained by multiplying the previous term by a constant value. For instance, in the sequence $3, \, 6, \, 12, \, 24...$ each number is multiplied by $2$ to get the next. This is a pattern with a common ratio (the quotient of a term and its previous term) of $2$.
repeating patterns—These patterns involve repeating elements in a regular interval. For example, in the sequence $\text{ABABAB}$, the letters $\text{A}$ and $\text{B}$ alternate in a regular cycle.
Fibonacci sequence—The Fibonacci sequence follows a specific pattern in which each term is the sum of the two preceding ones, starting from $0$ and $1$. The sequence begins as $0, \, 1, \, 1, \, 2, \, 3, \, 5, \, 8…$ and continues indefinitely. This sequence is frequently seen in nature.
alternating patterns—In an alternating pattern, the values or objects switch between two (or more) options in a predictable fashion. For example, $1, \, -1, \, 2, \, -2$ is a sequence with an alternating pattern in which the sign alternates between positive and negative.

Finding Missing Terms in a Sequence

To find missing terms in a sequence, you need to identify the type of pattern and the rule that defines it. Here are the steps:

Identify the pattern.—Determine whether it is an arithmetic, geometric, or other type of pattern.
Find the rule.—Look for the common difference (in arithmetic sequences), the common ratio (in geometric sequences), or another rule that defines the pattern.
Apply the rule.—Use the identified rule to fill in the missing term(s) of the sequence by continuing the pattern.

For example, if you have the sequence $5, \,$ ____ $, \, 13, \,$ ____ $, \, 21,$ and you need to find the missing terms, your procedure will look like this:

You will determine it is an arithmetic sequence because the difference between consecutive terms is constant.
You find the common difference is $4$, since $13 - 5 = 8$ and $21 - 13 = 8$, and the common difference between terms across two numbers is $8$, which means the common distance between consecutive terms is $\frac{8}{2} = 4$.
You find the missing terms by subtracting $4$ from $13$ to get $9$ and subtracting $4$ from $21$ to get $17$.

Thus, the missing terms are $9$ and $17$.

Probability

Probability deals with the likelihood or chance of a specific event occurring. It helps quantify uncertainty by assigning a numerical value between $0$ and $1$ to an event. Probability is used in a variety of fields to make predictions and decisions based on potential outcomes.

In probability, events can range from impossible to certain. A probability of $0$ means that the event will not occur, while a probability of $1$ means the event will definitely happen. Most probabilities fall somewhere between these two extremes, indicating varying degrees of likelihood. We write the probability of an event as such:

If $P(E) = 0$, the event is impossible and will not happen.
If $P(E) = 1$, the event is certain to happen.
If $P(E)$ falls between $0$ and $1$, that reflects the level of certainty of the event occurring. For example, a probability of $0.5$ (or $50\%$) means that the event is equally likely to happen or not happen.

Probability Terms

To understand probability in depth, it’s important to familiarize yourself with some key terms:

An experiment in probability is a procedure or action that produces an outcome. The outcome is not known in advance but can be one of several possibilities. For example, flipping a coin is an experiment in which the outcome is either heads or tails. Each flip represents an independent trial of the experiment.
An outcome refers to the result of a single trial of an experiment. Each possible result of an experiment is called an outcome. For example, when rolling a die, each number ($1, \, 2,\, 3,\, 4,\, 5,$ or $6$) is a possible outcome.
An event is a collection of one or more outcomes from an experiment. Events are subsets of the sample space, which includes all possible outcomes. For example, rolling an odd number on a die ($1, \, 3,$ or $5$) is an event, as is rolling a number greater than $4$ ($5$ or $6$).
The sample space is the set of all possible outcomes in a probability experiment. It provides the context within which probabilities are calculated. For example, for a six-sided die, the sample space is $\{1, \, 2, \, 3, \, 4,\, 5,\, 6 \}$.
Events are said to be independent if the occurrence or non-occurrence of one event does not affect the probability of another. For example, when flipping a coin twice, the result of the first toss does not affect the result of the second toss. These are independent events.

Calculating Probability

The probability of an event is calculated using the following formula:

\[P(E) = \frac{fo}{po}\]

where $P(E)$ is the probability of the event ($E$), $fo$ is the number of favorable outcomes (the outcomes in the sample space that satisfy the event), and $po$ is the total number of possible outcomes in the sample space.

For example, suppose you want to calculate the probability of rolling a $2$ on a standard six-sided die (an event we’ll designate as $T$). Since there is only one favorable outcome (rolling a $2$) out of six possible outcomes, the probability is $\frac{1}{6}$, which is approximately $0.167$ or $16.7\%$. That formula for this event is:

\[P(T) = \frac{1}{6}\]

Now, let’s say you want to interpret the probability of flipping a coin and getting heads ($H$). A coin has two equally likely outcomes: heads and tails. So, the probability of getting heads is calculated as:

\[P(H) = \frac{1}{2} = 0.5\]

This means the chance of getting heads is $0.5$, or $50\%$. If you flip the coin $10$ times, you would expect to get heads about five times, although the actual result may vary due to randomness.

Let’s look at another example.

Consider a standard deck of $52$ playing cards, where $26$ cards are red (hearts and diamonds). If you want to find the probability of drawing a red card ($R$), you’d write the formula as such:

\[P(R) = \frac{26}{52} = \frac{1}{2} = 0.5\]

This means there is a $50%$ chance of drawing a red card from a standard deck.

Now, let’s say you’re asked to find the probability of drawing a king ($K$) from the deck. There are four kings in a deck of $52$ cards:

\[P(K) = \frac{4}{52} = \frac{1}{13} \approx 0.077\]

This means there is approximately a $7.7\%$ chance of drawing a king from the deck.

In both examples, probability helps us quantify how likely certain events are based on the structure of the experiment, which allows us to make informed predictions.

Properties of Numbers

In mathematics, the properties of numbers form the foundation for solving everything from simple arithmetic problems to more complex algebraic equations. Understanding these properties allows us to simplify expressions, solve equations, and make logical connections between numbers. These properties also help us recognize patterns and determine solutions efficiently in calculations and problem-solving contexts. Here are some more number concepts that hold true.

Prime and Composite Numbers

Prime numbers are defined as numbers greater than $1$ that have no divisors other than $1$ and themselves. In other words, a prime number can only be divided evenly by $1$ and the number itself, with no other factors. Common examples of prime numbers include $2, \, 3, \, 5, \, 7, \, 11,$ and $13$. Notably, $2$ is the only even prime number.

Numbers that have more than two divisors are called composite numbers. For instance, $4, \, 6,$ and $8$ are composite because they can be divided by more than just $1$ and themselves.

Even and Odd Integers

Integers are whole numbers that can be positive, negative, or zero. Even integers are numbers that can be divided by $2$ without leaving a remainder, such as $-4, \,0, \, 2,$ and $6$. Odd integers, on the other hand, cannot be evenly divided by $2$, and examples include $-3, \,1, \,5,$ and $7$.

These are some important rules of integers:

The sum of two even numbers or two odd numbers is always even.
The sum of an even and an odd number is always odd.
Multiplying two even numbers results in an even number, while multiplying two odd numbers results in an odd number.
Multiplying an even number by an odd number results in an even number.
The division of two integers does not always result in an integer. For example, dividing $4$ by $2$ gives an integer ($2$), but dividing $3$ by $2$ results in a fraction $\left( \frac{3}{2} \right)$, not an integer.

Positive and Negative Integers

Positive integers are numbers greater than zero, such as $1, \,2,$ and $3$, while negative integers are numbers less than zero, such as $-1, \,-2,$ and $-3$. Zero is neither positive nor negative. Positive and negative integers follow specific rules in arithmetic:

The sum of two positive integers is always positive.
The sum of two negative integers is always negative.
The sum of a positive and a negative integer depends on their absolute values; the result takes the sign of the larger absolute value.
Multiplying two integers of the same sign results in a positive product, while multiplying integers of different signs results in a negative product.
Dividing two integers of the same sign results in a positive quotient, while dividing integers of different signs results in a negative quotient.