Math

Laws of Probability: Formulas, Examples and Probability Rules

5 minutes ago
11 mins
1 Views
Learn the laws of probability with simple formulas, clear explanations, solved examples, practice questions, and an interactive calculator. Perfect for IGCSE, GCSE, IB, AP, and high school math students.
Mathematics · Probability · High School

Laws of Probability

Master the fundamental rules that govern probability — with clear formulas, step-by-step examples, and exam-ready practice for GCSE, IGCSE, IB, AP, and the American curriculum.

GCSE IGCSE IB AP Statistics American Curriculum

What are the Laws of Probability?

The laws of probability are a set of mathematical rules that define how the likelihood of events is measured, combined, and compared. They establish that every probability is a number between 0 (impossible) and 1 (certain), and they provide the addition, multiplication, complement, and conditional rules used to calculate the probability of combined or related events.

💡 Why it matters: Every branch of mathematics that deals with uncertainty — from statistics to machine learning — is built on these foundational rules.

Key Concepts of the Laws of Probability

Before applying the laws, you need to understand the core vocabulary. Each concept below is used directly in the probability rules.

🎯

Probability Scale (0 to 1)

A probability is always a number from 0 (will never happen) to 1 (will definitely happen). A value of 0.5 means a 50% chance. You can also express probabilities as fractions or percentages.

✖️

Impossible & Certain Events

An impossible event has probability 0 — for example, rolling a 7 on a standard die. A certain event has probability 1 — for example, getting a number between 1 and 6 when rolling a die.

🔄

Complementary Events

The complement of event A (written A') is the event that A does not happen. Because every trial must result in either A or A', these two probabilities always add to exactly 1.

⚡

Mutually Exclusive Events

Two events are mutually exclusive if they cannot both happen at the same time. Rolling a 2 and rolling a 5 on one die are mutually exclusive — you can only get one result per roll.

🎲

Independent Events

Two events are independent if the outcome of one has absolutely no effect on the probability of the other. Flipping a coin twice is a classic example — the first flip does not influence the second.

🔗

Dependent Events

Dependent events are linked — the outcome of the first affects the probability of the second. Drawing two cards without replacement is a classic example, because the deck changes after the first draw.

📋

Sample Space

The sample space (S) is the complete set of all possible outcomes of an experiment. When rolling a fair die, S = {1, 2, 3, 4, 5, 6}. The probability of the entire sample space is always 1.

✅

Favorable Outcomes

Favorable outcomes are the specific outcomes that match the event you are calculating. The basic probability formula is: P(A) = (Number of favorable outcomes) ÷ (Total number of outcomes in S).

Laws of Probability Formulas

Here are all the core probability laws formulas you need to know. Each formula is followed by a plain-English explanation so you can understand exactly what it means.

0 ≤ P(A) ≤ 1
Probability Bounds Rule

Every probability must be at least 0 and at most 1. A value below 0 or above 1 is always wrong.

P(S) = 1
Certain Event / Total Probability

The probability of the entire sample space is 1. Something from the set of all possible outcomes must happen.

P(∅) = 0
Impossible Event Rule

The probability of an impossible event (the empty set) is 0. If an event cannot occur, its probability is zero.

P(A') = 1 − P(A)
Complement Rule

The probability that A does NOT happen equals 1 minus the probability that it does. Very useful when "not happening" is easier to calculate.

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
General Addition Rule

The probability that A or B occurs. You subtract P(A ∩ B) to avoid counting the overlap twice.

P(A ∪ B) = P(A) + P(B)
(if A and B are mutually exclusive)
Addition Rule — Mutually Exclusive

When A and B cannot both happen, there is no overlap, so P(A ∩ B) = 0 and the formula simplifies to simple addition.

P(A ∩ B) = P(A) × P(B)
(if A and B are independent)
Multiplication Rule — Independent Events

When two events are independent, the probability that both occur is found by multiplying their individual probabilities.

P(A|B) = P(A ∩ B) / P(B)
Conditional Probability Rule

The probability of A given that B has already occurred. You restrict the sample space to B and then check how likely A is within that restricted space.

How the Laws of Probability Work

Knowing the formulas is one thing — knowing when to apply each rule is what actually gets you marks. Here is a clear breakdown of each rule and the situations that call for it.

1. Complement Rule — "Find the probability it does NOT happen"

Use when: The question asks for the probability of an event NOT occurring, or when calculating the direct probability is more complex than calculating its complement.

How it works: The probability of everything adds up to 1. If you know P(A), then what is left over must be P(A'). Simply subtract from 1.

P(A') = 1 − P(A)

Example trigger phrase: "What is the probability of NOT getting a red card?"

2. Addition Rule — "Find the probability of A or B"

Use when: The question uses the words "or", "at least one of", or "either ... or ...".

How it works: Add the two individual probabilities, then subtract the overlap (events that belong to both A and B) because those outcomes were counted twice. If A and B are mutually exclusive (no overlap), skip the subtraction.

General: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Mutually Exclusive: P(A ∪ B) = P(A) + P(B)

Example trigger phrase: "What is the probability of drawing a heart or a face card?"

3. Multiplication Rule — "Find the probability of A and B both happening"

Use when: The question uses the words "and", "both", or asks for the probability of a sequence of events all occurring.

How it works: For independent events, simply multiply the two probabilities. For dependent events, multiply the probability of the first event by the conditional probability of the second event given the first already happened.

Independent: P(A ∩ B) = P(A) × P(B)
Dependent: P(A ∩ B) = P(A) × P(B|A)

Example trigger phrase: "What is the probability of getting heads on both flips?"

4. Conditional Probability Rule — "Find the probability given something already happened"

Use when: The question contains the word "given that", "if", or when the sample space has already been narrowed down by a known result.

How it works: You are working within a restricted sample space — only outcomes where B is true. Divide the probability of both A and B occurring by the probability of B.

P(A|B) = P(A ∩ B) / P(B)

Example trigger phrase: "Given that the card drawn is red, what is the probability it is a heart?"

Laws of Probability — Worked Examples

Study these step-by-step examples carefully. Each one uses a different probability rule so you can see them all in action.

Example 1 Complement Rule · Dice

❓ Problem: A fair six-sided die is rolled once. What is the probability of NOT getting a 6?

Given

Die has 6 equally likely outcomes: {1, 2, 3, 4, 5, 6}
Event A = rolling a 6
P(A) = 1/6

Rule Used

P(A') = 1 − P(A)

Step-by-Step Solution

Step 1: Identify P(A) — the probability of the event we want the complement of.

P(rolling a 6) = 1/6

Step 2: Apply the complement rule.

P(not a 6) = 1 − 1/6 = 5/6

✅ Final Answer: P(not a 6) = 5/6 ≈ 0.833

There are 5 outcomes that are not a 6 out of 6 possible outcomes, confirming the answer.

Example 2 Addition Rule (with overlap) · Cards

❓ Problem: A card is drawn at random from a standard 52-card deck. What is the probability that the card is a heart or a king?

Given

Total cards = 52
Hearts = 13 → P(H) = 13/52
Kings = 4 → P(K) = 4/52
King of Hearts = 1 → P(H ∩ K) = 1/52

Rule Used

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Step-by-Step Solution

Step 1: P(heart) = 13/52 = 1/4

Step 2: P(king) = 4/52 = 1/13

Step 3: P(heart AND king) = P(king of hearts) = 1/52

Step 4: Apply the addition rule to avoid double-counting the king of hearts:

P(H ∪ K) = 13/52 + 4/52 − 1/52 = 16/52 = 4/13

✅ Final Answer: P(heart or king) = 16/52 = 4/13 ≈ 0.308

There are 16 cards that are either a heart or a king (13 hearts + 3 non-heart kings). We must subtract the overlap because the king of hearts is both a heart and a king.

Example 3 Multiplication Rule — Independent Events · Coins

❓ Problem: A fair coin is flipped twice. What is the probability of getting heads on both flips?

Given

P(H on flip 1) = 1/2
P(H on flip 2) = 1/2
The two flips are independent

Rule Used

P(A ∩ B) = P(A) × P(B)

Step-by-Step Solution

Step 1: Confirm independence. The result of the first flip has no effect on the second. ✓

Step 2: Apply the multiplication rule for independent events:

P(HH) = P(H) × P(H) = 1/2 × 1/2 = 1/4

✅ Final Answer: P(two heads) = 1/4 = 0.25

The sample space has 4 outcomes: {HH, HT, TH, TT}. Only one outcome gives two heads, confirming P = 1/4.

Example 4 Addition Rule — Mutually Exclusive · Classroom Survey

❓ Problem: In a class of 30 students, 12 study French and 8 study Spanish. No student studies both. If a student is chosen at random, what is the probability they study French or Spanish?

Given

Total students = 30
P(French) = 12/30 = 2/5
P(Spanish) = 8/30 = 4/15
Mutually exclusive: P(F ∩ S) = 0

Rule Used

P(A ∪ B) = P(A) + P(B)

Step-by-Step Solution

Step 1: Since no student studies both languages, the events are mutually exclusive, so P(F ∩ S) = 0.

Step 2: Apply the simplified addition rule:

P(F ∪ S) = 12/30 + 8/30 = 20/30 = 2/3

✅ Final Answer: P(French or Spanish) = 2/3 ≈ 0.667

20 out of 30 students study one of the two languages — that is exactly 2/3 of the class.

Example 5 Conditional Probability · Cards

❓ Problem: A card is drawn from a standard deck. Given that the card is a red card, what is the probability that it is a heart?

Given

Event A = the card is a heart
Event B = the card is red
P(A ∩ B) = P(heart) = 13/52
P(B) = P(red) = 26/52

Rule Used

P(A|B) = P(A ∩ B) / P(B)

Step-by-Step Solution

Step 1: All hearts are red cards, so P(heart AND red) = P(heart) = 13/52.

Step 2: P(red) = 26/52 (half the deck is red).

Step 3: Apply conditional probability:

P(heart | red) = (13/52) / (26/52) = 13/26 = 1/2

✅ Final Answer: P(heart | red) = 1/2 = 0.5

Once we know the card is red, our sample space reduces to the 26 red cards — exactly half of which (13) are hearts.

Real-World Applications

Probability laws are not just academic exercises. They underpin decision-making in almost every industry.

🌦️

Weather Forecasting

Meteorologists use conditional probability to issue forecasts like "70% chance of rain." This estimate is based on how often similar atmospheric patterns historically led to rainfall.

⚽

Games and Sports

Sports analysts and betting markets use probability to quantify a team's chance of winning a match. The multiplication rule helps calculate the probability of winning multiple games in a row.

🏦

Insurance and Risk

Insurers calculate premiums using the probability of events like accidents or illness. The complement rule is used to model the probability that a claim is NOT filed, and combined events are handled with the addition rule.

🏥

Medical Testing

Doctors use conditional probability (Bayes' theorem) to interpret test results. A positive test doesn't mean certain illness — the result must be weighed against the base rate of the disease in the population.

📊

Statistics and Data Science

Probability distributions, hypothesis testing, and confidence intervals are all built on the foundational probability laws. Data scientists rely on these rules every time they model uncertainty in data.

🤖

Machine Learning and AI

Machine learning models — including spam filters, recommendation engines, and medical AI — use probabilistic reasoning to make predictions. Naive Bayes classifiers directly apply the multiplication rule for independent events.

Common Mistakes Students Make

These are the mistakes that appear most often in exams. Knowing them in advance can save you marks.

❌ Mistake 1: Adding when you should multiply

If the question asks for the probability that two independent events BOTH occur, you must multiply, not add. P(A and B) = P(A) × P(B), not P(A) + P(B).

❌ Mistake 2: Multiplying when you should add

For "or" questions (the union), you add the probabilities and subtract the overlap. Multiplying will give you a much smaller number and a completely wrong answer.

❌ Mistake 3: Forgetting to subtract the overlap

When events are NOT mutually exclusive, you must subtract P(A ∩ B) in the addition rule. Skipping this step counts overlapping outcomes twice and inflates the answer.

❌ Mistake 4: Confusing mutually exclusive with independent

These are completely different concepts. Mutually exclusive events cannot both happen at the same time. Independent events do not influence each other's probability. In fact, if two events with non-zero probability are mutually exclusive, they CANNOT be independent.

❌ Mistake 5: Misusing the complement rule

The complement rule applies to a single event: P(A') = 1 − P(A). Students sometimes subtract the probability of the wrong event, or apply it to unions/intersections incorrectly.

❌ Mistake 6: Getting a probability outside [0, 1]

A probability can never be negative or greater than 1. If your answer is 1.3 or −0.2, you have made an error. Always check your answer against the bounds rule: 0 ≤ P(A) ≤ 1.

Practice Questions

Test your understanding with these exam-style questions. Try each one yourself before revealing the answer.

Question 1

A fair six-sided die is rolled. What is the probability of not rolling a 3?

Question 2

A card is drawn at random from a standard 52-card deck. What is the probability that the card is a queen or a spade?

Question 3

A fair coin is flipped twice. What is the probability of getting two tails?

Question 4

P(A) = 0.4 and P(B) = 0.5. A and B are independent events. Find P(A ∩ B).

Question 5

P(A) = 0.7. Find P(A').

Probability Laws Calculator

Use this interactive tool to apply the complement, addition, and multiplication rules instantly. Enter probabilities as decimals (e.g., 0.4 for 40%).

Formula: P(A') = 1 − P(A)

Summary — Key Takeaways

  • Every probability is a number between 0 (impossible) and 1 (certain), and the total probability of all outcomes in the sample space always equals 1.
  • The complement rule P(A') = 1 − P(A) is used when it is easier to find the probability of an event NOT happening.
  • The addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B) gives the probability of A or B. Subtract the overlap to avoid double-counting.
  • If events are mutually exclusive, P(A ∩ B) = 0, and the addition rule simplifies to P(A ∪ B) = P(A) + P(B).
  • The multiplication rule for independent events: P(A ∩ B) = P(A) × P(B). Use it when both events must happen together.
  • Conditional probability P(A|B) = P(A ∩ B) / P(B) applies when the sample space has been restricted by a known outcome.
  • Mutually exclusive means the events cannot occur at the same time. Independent means one event does not affect the other's probability. Do not confuse them.
  • Always verify your final answer sits within [0, 1] — an answer outside this range means there is a calculation error.

Frequently Asked Questions

📘 RevisionTown — Helping students master mathematics

GCSE · IGCSE · IB · AP · American Curriculum  |  Laws of Probability Study Guide

Shares: