Probability Basics: Rules, Formulas, and Practice Problems

2026.04.14

Learn.byAuthor

What are the chances? Probability is the math of uncertainty — it tells you how likely an event is to happen, from coin flips to real-world predictions.

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Key Point: Probability is tested on AP Statistics, SAT, ACT, and GCSE exams. The core rules are simple, but applying them correctly under exam pressure takes practice.

What Is Probability?

Definition and Basic Formula

Probability measures how likely an event is to occur, expressed as a number between 0 and 1.

P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}

$P(A) = 0$ : Impossible event
$P(A) = 1$ : Certain event
$0 < P(A) < 1$ : Somewhere in between

Key Probability Rules

Rule	Formula	When to Use
Complement	$P(A') = 1 - P(A)$	"Not A" probability
Addition (Mutually Exclusive)	$P(A \cup B) = P(A) + P(B)$	A and B can’t happen together
Addition (General)	$P(A \cup B) = P(A) + P(B) - P(A \cap B)$	A and B can overlap
Multiplication (Independent)	$P(A \cap B) = P(A) \times P(B)$	A doesn’t affect B
Conditional	$P(A\|B) = \frac{P(A \cap B)}{P(B)}$	Probability of A given B

Types of Events

Independent Events

Two events are independent if one doesn’t affect the other.

Rolling a die twice: each roll is independent.
$P(A \cap B) = P(A) \times P(B)$

Mutually Exclusive Events

Two events are mutually exclusive if they can’t happen at the same time.

Drawing a card that is both a King and a Queen: impossible.
$P(A \cap B) = 0$

Conditional Probability

The probability of A happening given that B has already happened.

P(A|B) = \frac{P(A \cap B)}{P(B)}

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Watch Out: Don’t confuse

P(A \cap B)

with

P(A|B)

. The intersection is a joint probability, while conditional probability assumes B has already occurred.

Counting Methods

Permutations (Order Matters)

P(n, r) = \frac{n!}{(n-r)!}

Combinations (Order Doesn’t Matter)

C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}

Worked Examples

Example 1: Basic Probability

A bag has 3 red, 5 blue, and 2 green balls. What’s the probability of drawing a blue ball?

Show Solution

Total balls: $3 + 5 + 2 = 10$

P(\text{blue}) = \frac{5}{10} = \frac{1}{2}

Example 2: Addition Rule

A card is drawn from a standard deck. What’s the probability it’s a King or a Heart?

Show Solution

$P(\text{King}) = \frac{4}{52}$ , $P(\text{Heart}) = \frac{13}{52}$ , $P(\text{King} \cap \text{Heart}) = \frac{1}{52}$

P(\text{King} \cup \text{Heart}) = \frac{4}{52} + \frac{13}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}

Example 3: Conditional Probability

In a class of 30 students, 18 study math and 12 study both math and science. If a student studies math, what’s the probability they also study science?

Show Solution

P(\text{Science}|\text{Math}) = \frac{12}{18} = \frac{2}{3}

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Pro Tip: When stuck on a probability problem, draw a tree diagram or Venn diagram. Visual representations make it much easier to identify the correct formula to use.

Top 3 Common Mistakes

Not subtracting the overlap in the addition rule — if events can happen together, you must subtract $P(A \cap B)$
Confusing independent with mutually exclusive — mutually exclusive events are NOT independent (if one happens, the other can’t)
Using the wrong denominator for conditional probability — the condition restricts the sample space

Probability Basics: Rules, Formulas, and Practice Problems

What Is Probability?

Definition and Basic Formula

Key Probability Rules

Types of Events

Independent Events

Mutually Exclusive Events

Conditional Probability

Counting Methods

Permutations (Order Matters)

Combinations (Order Doesn’t Matter)

Worked Examples

Example 1: Basic Probability

Example 2: Addition Rule

Example 3: Conditional Probability

Top 3 Common Mistakes

Related Topics