What is the basic probability formula?

The basic probability formula is P(E) = Number of Favorable Outcomes ÷ Total Number of Outcomes. It gives a value between 0 and 1, where 0 means the event is impossible and 1 means the event is certain.

What is the addition rule in probability?

The addition rule states P(A or B) = P(A) + P(B) − P(A and B). For mutually exclusive events, P(A and B) = 0, so it simplifies to P(A or B) = P(A) + P(B).

What is the difference between independent and dependent events?

Independent events do not affect each other's probability. For example, flipping a coin and rolling a die are independent. Dependent events change the probability of subsequent events, such as drawing cards from a deck without replacement.

How do you calculate conditional probability?

Conditional probability is calculated using the formula P(A|B) = P(A and B) ÷ P(B). It finds the probability of event A occurring given that event B has already occurred.

What is the complement rule in probability?

The complement rule states that P(not A) = 1 − P(A). It is useful when it is easier to calculate the probability of an event not happening and then subtract from 1.

What are mutually exclusive events?

Mutually exclusive events are events that cannot occur at the same time. For example, when rolling a die, getting a 2 and getting a 5 on the same roll are mutually exclusive. If A and B are mutually exclusive, P(A and B) = 0.

How is probability used in real life?

Probability is used in weather forecasting, sports analytics, insurance and finance (risk assessment), medical testing (sensitivity and specificity), games and gambling, business decision-making, and many scientific fields.

Probability Formulas – Complete Study Guide

Last updated: March 2026 | AP · IB · GCSE · IGCSE · SAT

1. Introduction to Probability

What Is Probability?

Probability is the branch of mathematics that deals with measuring how likely an event is to occur. Think of it as a numerical way of answering the everyday question, "What are the chances?" Every time you check a weather forecast, wonder whether your favourite team will win, or decide whether to carry an umbrella, you are — perhaps without realising it — thinking about probability.

In formal terms, probability assigns a number between 0 and 1 to an event:

0 means the event is impossible — it can never happen. For example, the probability of rolling a 7 on a standard six-sided die is 0.
1 means the event is certain — it will definitely happen. For example, the probability of getting heads or tails when you flip a fair coin is 1 (one of the two must happen).
Any value between 0 and 1 represents varying degrees of likelihood. The closer the number is to 1, the more likely the event; the closer to 0, the less likely.

You can also express probability as a percentage (multiply by 100) or as a fraction. For instance, a probability of 0.25 is the same as 25 % or ¼.

Why Is Probability Important?

Probability is not just an abstract mathematical idea that lives inside textbooks. It is one of the most practical areas of mathematics because it helps us make better decisions under uncertainty. Here is why it matters:

Science and medicine: Researchers use probability to design experiments, interpret data, and determine whether a new drug is effective.
Business and finance: Companies model risk, forecast sales, and set insurance premiums using probability.
Technology: Machine-learning algorithms, spam filters, and recommendation engines all rely on probabilistic models.
Everyday life: Whether you are deciding to buy a lottery ticket or choosing the fastest route to school, you are implicitly estimating probabilities.

The Language of Probability

Before we dive into formulas, let us nail down the vocabulary you will see throughout this guide — and on every exam.

Experiment (or Trial): Any process that produces a result. Rolling a die, flipping a coin, drawing a card — each of these is an experiment.
Outcome: A single possible result of an experiment. When you roll a die, one outcome is "getting a 4."
Sample Space (S): The complete set of all possible outcomes. For a coin flip, S = {Heads, Tails}. For a die roll, S = {1, 2, 3, 4, 5, 6}.
Event (E): A specific outcome or a group of outcomes that we are interested in. "Rolling an even number" is an event that contains the outcomes {2, 4, 6}.
Favorable Outcomes: The outcomes within the sample space that satisfy the event we care about.

💡 Teacher Tip: Always start a probability problem by writing out the sample space. If you know the total number of outcomes and the favorable outcomes, you can solve almost any basic probability question.

Impossible vs. Certain Events — and Everything in Between

Imagine a bag that contains 5 red marbles and nothing else.

The probability of drawing a red marble = 5/5 = 1 (certain event).
The probability of drawing a blue marble = 0/5 = 0 (impossible event).

Most real-world events fall somewhere between these two extremes. That is what makes probability both interesting and useful — it helps us quantify the "grey area."

🌧️ Real-Life Example — Weather Forecast

When a weather app says there is a 70 % chance of rain, it means the probability of rain is 0.70. Out of many days with similar conditions, about 70 out of 100 would see rain. It does not mean it will rain for 70 % of the day!

2. The Basic Probability Formula

This is the single most important formula in probability and the foundation for everything else you will learn. Memorise it and understand it deeply.

P(E) = Number of Favorable Outcomes ÷ Total Number of Outcomes

Breaking Down the Formula

P(E) — reads "the probability of event E." It is the value we want to find.
Number of Favorable Outcomes — how many outcomes in the sample space satisfy the condition described by event E.
Total Number of Outcomes — the size of the sample space (every possible outcome of the experiment).

Why does this formula work? It works because, in a fair experiment where every outcome is equally likely, the chance of a specific event is simply the proportion of outcomes that satisfy the event. If 3 out of 6 faces of a die are even, the proportion is 3/6 = 1/2. That is a 50 % chance.

⚠️ Important Condition: This formula assumes that all outcomes are equally likely. It does not apply directly when outcomes have different probabilities (for example, a loaded die).

Example 1 — Coin Toss

Question: What is the probability of getting Heads when you flip a fair coin?

Step 1 – Identify the sample space: S = {Heads, Tails}. Total outcomes = 2.

Step 2 – Identify favorable outcomes: We want Heads. Favorable outcomes = {Heads} = 1.

Step 3 – Apply the formula:

P(Heads) = 1 / 2 = 0.5 = 50 %

Interpretation: If you flip a fair coin many times, you would expect Heads roughly half the time.

Example 2 — Rolling a Die

Question: What is the probability of rolling a 3 on a standard six-sided die?

Step 1 – Sample space: S = {1, 2, 3, 4, 5, 6}. Total outcomes = 6.

Step 2 – Favorable outcomes: Event = {3}. Favorable = 1.

Step 3 – Apply formula:

P(3) = 1 / 6 ≈ 0.1667 ≈ 16.67 %

Interpretation: Each face of a fair die has an equal 1-in-6 chance of landing face up.

Example 3 — Drawing a Card

Question: A standard deck has 52 cards. What is the probability of drawing an Ace?

Step 1 – Total outcomes: 52 (one for each card in the deck).

Step 2 – Favorable outcomes: There are 4 Aces (♠ ♥ ♦ ♣). Favorable = 4.

Step 3 – Apply formula:

P(Ace) = 4 / 52 = 1 / 13 ≈ 0.0769 ≈ 7.69 %

Interpretation: Roughly 1 in every 13 cards you draw at random will be an Ace.

3. Key Probability Concepts

Before tackling the more advanced formulas, you need a rock-solid understanding of several foundational concepts. These ideas come up in every probability question, and examiners love testing whether you truly understand the difference between them.

3.1 Sample Space

The sample space is the set of all possible outcomes. We typically label it S. Getting the sample space right is the critical first step for every probability problem.

Sample Space Examples

Coin flip: S = {H, T} — 2 outcomes.
Rolling a die: S = {1, 2, 3, 4, 5, 6} — 6 outcomes.
Two coin flips: S = {HH, HT, TH, TT} — 4 outcomes. Notice that HT and TH are different outcomes because the order matters (first flip vs. second flip).
Rolling two dice: S has 6 × 6 = 36 outcomes (each die contributes 6 possibilities).

3.2 Events

An event is any subset of the sample space. An event can contain one outcome, several outcomes, or even every outcome in the sample space. We typically use capital letters (A, B, E, etc.) to name events.

Simple event: Contains exactly one outcome. Example: Rolling a 4.
Compound event: Contains more than one outcome. Example: Rolling an even number = {2, 4, 6}.

3.3 Mutually Exclusive Events

Two events are mutually exclusive (also called disjoint) if they cannot happen at the same time. If one occurs, the other is automatically ruled out.

If A and B are mutually exclusive: P(A and B) = 0

Example – Mutually Exclusive Events

When rolling a single die:

Event A = rolling a 2
Event B = rolling a 5

You cannot roll a 2 and a 5 at the same time on a single die, so A and B are mutually exclusive. P(A and B) = 0.

Counter-example: Event C = rolling an even number {2, 4, 6}; Event D = rolling a number greater than 3 {4, 5, 6}. These are not mutually exclusive because 4 and 6 appear in both events — they can occur simultaneously.

3.4 Independent Events

Two events are independent if the occurrence of one does not affect the probability of the other. Knowing that one event happened gives you zero extra information about whether the other will happen.

If A and B are independent: P(A and B) = P(A) × P(B)

Example – Independent Events

You flip a coin and roll a die. Let A = coin lands Heads, B = die shows 6.

The result of the coin does not influence the die, so A and B are independent.
P(A) = 1/2, P(B) = 1/6.
P(A and B) = (1/2) × (1/6) = 1/12 ≈ 8.33 %.

Intuition: Out of 12 equally likely coin-and-die combinations, exactly 1 is (Heads, 6).

3.5 Dependent Events

Two events are dependent if the occurrence of one changes the probability of the other. This often happens when objects are removed from a set and not replaced.

Example – Dependent Events (Drawing Without Replacement)

A bag contains 5 red balls and 3 blue balls. You draw two balls one after the other without replacement.

P(first ball is red) = 5/8.
If the first ball drawn was red, there are now 4 red and 3 blue balls left (7 balls total).
P(second ball is red | first was red) = 4/7.

The probability of the second draw depends on what happened in the first draw — these events are dependent.

P(both red) = (5/8) × (4/7) = 20/56 = 5/14 ≈ 0.357.

💡 Quick Test for Independence: Ask yourself: "Does knowing the result of Event A change the probability of Event B?" If yes → dependent. If no → independent.

4. Important Probability Rules

These rules are the toolkit you will reach for again and again. Each rule has a specific purpose, and knowing when to use each one is just as important as knowing the formula itself.

4.1 The Addition Rule (OR Rule)

Use the Addition Rule when you want the probability that Event A or Event B (or both) occurs.

P(A or B) = P(A) + P(B) − P(A and B)

Why subtract P(A and B)? Because when we add P(A) and P(B), the outcomes that belong to both A and B get counted twice — once in P(A) and once in P(B). Subtracting P(A and B) corrects for that double-counting.

Special case — Mutually Exclusive Events: If A and B cannot happen at the same time, then P(A and B) = 0, and the formula simplifies to:

P(A or B) = P(A) + P(B) [mutually exclusive events]

Solved Example – Addition Rule

Question: A card is drawn at random from a standard 52-card deck. What is the probability that the card is a King or a Heart?

Step 1 – Define events: A = card is a King; B = card is a Heart.

Step 2 – Find individual probabilities:

P(A) = 4/52 (there are 4 Kings).
P(B) = 13/52 (there are 13 Hearts).

Step 3 – Find P(A and B): The King of Hearts is both a King and a Heart, so P(A and B) = 1/52.

Step 4 – Apply the Addition Rule:

P(King or Heart) = 4/52 + 13/52 − 1/52 = 16/52 = 4/13 ≈ 0.3077 ≈ 30.77 %

Interpretation: About 31 % of cards in a deck are either a King or a Heart (or the King of Hearts which is both).

4.2 The Multiplication Rule (AND Rule)

Use the Multiplication Rule when you want the probability that Event A and Event B both occur.

P(A and B) = P(A) × P(B | A)

Here, P(B | A) is the probability of B given that A has already occurred. If A and B are independent, P(B | A) simply equals P(B), and the formula simplifies to P(A) × P(B).

Solved Example – Multiplication Rule (Independent Events)

Question: You flip a fair coin twice. What is the probability of getting Heads both times?

Step 1: A = first flip is Heads, B = second flip is Heads. The flips are independent.

Step 2: P(A) = 1/2, P(B) = 1/2.

Step 3:

P(HH) = (1/2) × (1/2) = 1/4 = 0.25 = 25 %

Solved Example – Multiplication Rule (Dependent Events)

Question: A jar has 6 green and 4 yellow sweets. You take two sweets without replacement. What is the probability both are green?

Step 1: P(first green) = 6/10 = 3/5.

Step 2: After removing one green, there are now 5 green and 4 yellow left (9 total). P(second green | first green) = 5/9.

Step 3:

P(both green) = (6/10) × (5/9) = 30/90 = 1/3 ≈ 0.333 ≈ 33.3 %

Interpretation: About one in three times you draw two sweets, both will be green.

4.3 The Complement Rule

The complement of event A, written as A′ (or Ā or A^c), is "everything that is not A." The complement rule is a powerful shortcut when calculating P(A) directly is hard but calculating the opposite is easy.

P(A′) = 1 − P(A) or equivalently P(A) = 1 − P(A′)

Why does it work? The total probability of all possible outcomes is always 1. If an event has probability P(A), then the probability of everything else must account for the rest, which is 1 − P(A).

Solved Example – Complement Rule

Question: A die is rolled. What is the probability of not rolling a 6?

Step 1: P(rolling a 6) = 1/6.

Step 2: Apply the complement rule:

P(not 6) = 1 − 1/6 = 5/6 ≈ 0.833 ≈ 83.3 %

When is this useful? If you need "at least one" of something over multiple trials, it is usually much easier to calculate the probability of "none" and subtract from 1.

Powerful Application – "At Least One" Problems

Question: You roll a die 3 times. What is the probability of getting at least one 6?

Step 1: P(not getting a 6 on one roll) = 5/6.

Step 2: P(no sixes in 3 rolls) = (5/6)³ = 125/216.

Step 3: P(at least one 6) = 1 − 125/216 = 91/216 ≈ 0.4213 ≈ 42.1 %

This complement approach saved us from having to individually calculate P(exactly one 6) + P(exactly two 6s) + P(three 6s).

4.4 Conditional Probability

Conditional probability answers the question: "What is the probability of event A given that we already know event B has occurred?" We write this as P(A | B) (read "probability of A given B").

P(A | B) = P(A and B) ÷ P(B) [provided P(B) > 0]

Intuition: Once we know B has happened, the sample space shrinks to only the outcomes inside B. We then look at what fraction of those outcomes also satisfy A.

Solved Example – Conditional Probability

Question: In a class of 30 students, 18 study Mathematics, 12 study Physics, and 6 study both. A student is chosen at random. Given that the student studies Mathematics, what is the probability they also study Physics?

Step 1: Let M = studies Maths, P = studies Physics.

P(M) = 18/30 = 3/5
P(M and P) = 6/30 = 1/5

Step 2: Apply the conditional probability formula:

P(P | M) = P(M and P) / P(M) = (1/5) / (3/5) = 1/3 ≈ 0.333 ≈ 33.3 %

Interpretation: Among students who study Maths, one-third also study Physics.

4.5 Independent Events Rule

We touched on independence earlier, but let us formalise it. Two events A and B are independent if and only if:

P(A and B) = P(A) × P(B)

This also means that P(A | B) = P(A) and P(B | A) = P(B) — knowing one happened does not change the other.

Solved Example – Testing for Independence

Question: A card is drawn from a deck. Let A = card is red, B = card is a Queen. Are A and B independent?

Step 1: P(A) = 26/52 = 1/2 (26 red cards). P(B) = 4/52 = 1/13 (4 Queens).

Step 2: P(A) × P(B) = (1/2)(1/13) = 1/26.

Step 3: P(A and B) = 2/52 = 1/26 (2 red Queens).

Step 4: Since P(A and B) = P(A) × P(B), A and B are independent. Being red tells you nothing about whether the card is a Queen (and vice versa), because Queens are evenly split between red and black.

5. Detailed Solved Examples

Let us work through a variety of problems step by step. Each example reinforces a different aspect of probability and mirrors the types of questions you will face on exams.

Example 1 — Rolling an Even Number on a Die

Question: A standard die is rolled once. What is the probability of getting an even number?

Step 1 – Total outcomes: S = {1, 2, 3, 4, 5, 6}. Total = 6.

Step 2 – Favorable outcomes: Even numbers = {2, 4, 6}. Favorable = 3.

Step 3 – Apply formula: P(even) = 3/6 = 1/2.

Step 4 – Simplify and interpret:

P(even) = 1/2 = 0.5 = 50 %

Half of the faces are even, so there is a 50 % chance of rolling an even number.

Example 2 — Drawing a Heart from a Deck

Question: What is the probability of drawing a Heart from a well-shuffled standard 52-card deck?

Step 1 – Total outcomes: 52 cards.

Step 2 – Favorable outcomes: 13 Hearts (Ace through King of Hearts).

Step 3 – Apply formula: P(Heart) = 13/52.

Step 4 – Simplify:

P(Heart) = 1/4 = 0.25 = 25 %

Since there are 4 suits of equal size, each suit has a 25 % probability.

Example 3 — Probability of Two Heads (Two Coin Flips)

Question: Two fair coins are flipped. What is the probability of getting Heads on both?

Step 1 – Total outcomes: S = {HH, HT, TH, TT}. Total = 4.

Step 2 – Favorable outcomes: {HH}. Favorable = 1.

Step 3 – Apply formula: P(HH) = 1/4.

Step 4 – Simplify:

P(both Heads) = 1/4 = 0.25 = 25 %

Alternative method (Multiplication Rule): P(H) × P(H) = (1/2)(1/2) = 1/4. Same answer!

Example 4 — Probability with Coloured Balls

Question: A bag contains 8 red, 5 blue, and 7 green balls. One ball is drawn at random. What is the probability it is blue?

Step 1 – Total outcomes: 8 + 5 + 7 = 20 balls.

Step 2 – Favorable outcomes: 5 blue balls.

Step 3 – Apply formula: P(blue) = 5/20.

Step 4 – Simplify:

P(blue) = 1/4 = 0.25 = 25 %

Follow-up: What is the probability it is not green? P(not green) = 1 − 7/20 = 13/20 = 0.65 = 65 %. (Complement Rule!)

Example 5 — Real-World Probability Scenario

Question: In a school, 60 % of students pass the Mathematics exam and 45 % pass the Science exam. If 30 % pass both, what is the probability a randomly selected student passes at least one of the two exams?

Step 1 – Define events: M = passes Maths, S = passes Science.

Step 2 – Given information: P(M) = 0.60, P(S) = 0.45, P(M and S) = 0.30.

Step 3 – Apply Addition Rule:

P(M or S) = P(M) + P(S) − P(M and S) = 0.60 + 0.45 − 0.30

P(at least one) = 0.75 = 75 %

Interpretation: Three-quarters of students pass at least one of the two exams. This means 25 % fail both.

6. Real-Life Applications of Probability

Understanding probability is not just about passing exams — it is a life skill. Here are some of the most important ways probability is used in the real world.

6.1 Weather Forecasting

Meteorologists collect huge amounts of atmospheric data — temperature, humidity, pressure, wind patterns — and feed it into probabilistic models. When you hear "there is a 40 % chance of rain tomorrow," the forecast is based on historical data showing that in similar atmospheric conditions, rain occurred approximately 40 % of the time. These forecasts rely on conditional probability (given these weather patterns, what is the probability of rain?) and Bayesian updating (continuously refining the prediction as new data comes in).

6.2 Sports Analytics

Modern sports teams use probability extensively. In baseball, the probability of a batter getting a hit against a specific pitcher is calculated using past performance data. In football (soccer), expected goals (xG) models assign a probability to every shot based on distance, angle, speed, and defensive pressure. Fantasy sports platforms and sports betting odds are built entirely on probability models. Coaches use probability to decide strategies — for example, whether to attempt a two-point conversion in American football or go for the field goal.

6.3 Insurance and Finance

Insurance companies are fundamentally probability businesses. They estimate the probability of events like car accidents, house fires, or health emergencies, and use these probabilities to set premiums. If the probability of a 25-year-old driver having an accident is 0.04 (4 %), the insurance company prices the policy to cover expected payouts plus a profit margin. In finance, probability is used to model stock price movements, assess portfolio risk, and price options using models like Black-Scholes, which is rooted in probability theory.

6.4 Medical Testing

Probability is critical in medical diagnostics. Every diagnostic test has:

Sensitivity — the probability that a test correctly identifies a person who has a disease (true positive rate).
Specificity — the probability that a test correctly identifies a person who does not have a disease (true negative rate).

Doctors use conditional probability and Bayes' Theorem to answer questions like: "Given that a patient tested positive, what is the actual probability they have the disease?" This is crucial because even highly accurate tests can produce misleading results when the disease is rare — a concept known as the base rate fallacy.

6.5 Games and Gambling

Games of chance — card games, dice games, lotteries, roulette — are the most direct applications of probability. Understanding probability helps you see why the "house always wins" in the long run (the concept of expected value). For example, in a fair lottery where you pick 6 numbers out of 49, the probability of winning the jackpot is 1 in 13,983,816. Casinos design every game so that the expected payout to players is less than the amount wagered, ensuring a profit over thousands of plays. Understanding probability can help you make informed decisions about when and how much to wager.

7. Common Student Mistakes

After teaching probability for over 20 years, I have seen the same mistakes come up again and again. Let us go through them so you can avoid losing marks on exam day.

❌ Mistake 1: Counting Outcomes Incorrectly

The error: Students often forget to count all possible outcomes, especially in compound experiments. For example, when flipping two coins, some students list only {HH, HT, TT} — forgetting TH. They confuse "one head and one tail" as a single outcome instead of recognising that HT (Heads first, Tails second) and TH (Tails first, Heads second) are two distinct outcomes.

The fix: Use a systematic listing method, a tree diagram, or a two-way table. For two dice, draw a 6 × 6 grid. For two coins, list everything explicitly: {HH, HT, TH, TT}. Systematic counting prevents missed outcomes.

❌ Mistake 2: Misunderstanding Independence

The error: After flipping Heads five times in a row, students often say "Tails must be due" — as if the coin has a memory. This is the Gambler's Fallacy. Each coin flip is independent; the coin does not "know" what happened before.

The fix: Remember that independent events have no memory. The probability of Heads on the sixth flip is still exactly 1/2, regardless of previous results. Independence means past outcomes do not influence future outcomes.

❌ Mistake 3: Confusing Mutually Exclusive and Independent

The error: Students often use "mutually exclusive" and "independent" interchangeably. They are very different concepts!

The fix:

Mutually exclusive = cannot happen at the same time → P(A and B) = 0.
Independent = one does not affect the other → P(A and B) = P(A) × P(B).

Key insight: If two events (each with non-zero probability) are mutually exclusive, they cannot be independent (because knowing one happened tells you the other definitely did not happen — that is a change in probability!). Conversely, independent events with non-zero probabilities are never mutually exclusive.

❌ Mistake 4: Misusing the Addition Rule (Forgetting to Subtract the Overlap)

The error: Students write P(A or B) = P(A) + P(B) without checking whether A and B overlap. This only works if A and B are mutually exclusive.

The fix: Always use the full formula P(A or B) = P(A) + P(B) − P(A and B). If the events are mutually exclusive, P(A and B) = 0, so it reduces naturally. Using the full formula every time prevents mistakes.

❌ Mistake 5: Not Adjusting for "Without Replacement"

The error: In problems involving drawing objects without replacement, students forget that the total number of items decreases after each draw. They use the original total for every probability calculation.

The fix: After each draw, reduce the total by 1 and adjust the number of favorable outcomes based on what was already drawn. If you drew a red ball first, there is one fewer red ball and one fewer ball overall.

8. Exam Tips and Problem-Solving Strategies

Probability questions are among the most scored-on topics in AP Statistics, IB Mathematics, GCSE, and IGCSE exams. Here is a step-by-step method to approach them confidently.

🎯 Strategy 1: Always Start by Defining the Sample Space

Before doing any calculation, write down all possible outcomes. For compound experiments, use tree diagrams, two-way tables, or systematic lists. Knowing the sample space helps you avoid miscounting, which is the number one source of errors.

🎯 Strategy 2: Identify What Type of Problem It Is

Ask yourself these questions in order:

Am I looking for "A or B"? → Addition Rule.
Am I looking for "A and B"? → Multiplication Rule.
Am I looking for "not A" or "at least one"? → Complement Rule.
Am I told that something has already happened? → Conditional Probability.
Are the events independent or dependent? → This determines which version of the multiplication rule to use.

🎯 Strategy 3: Draw It Out

Visual tools such as Venn diagrams, tree diagrams, and probability tables make problems much easier to understand and solve. Tree diagrams are especially useful for multi-step experiments such as drawing balls without replacement or sequential coin flips. Each branch represents a possible outcome, and you multiply along branches, then add across branches to find totals.

🎯 Strategy 4: Check Your Answer Makes Sense

After calculating, always verify:

Is your probability between 0 and 1? If not, there is an error.
Does the probability feel right? If you calculated a 90 % chance of a rare event, double-check.
Do all probabilities in the sample space sum to 1? If you calculated probabilities for all outcomes, they should total 1.

🎯 Strategy 5: Use the Complement for "At Least One" Problems

Whenever a problem asks for the probability of "at least one" occurrence of an event, calculate the probability of "zero occurrences" and subtract from 1. This is almost always quicker and less error-prone than listing every case individually.

🎯 Strategy 6: Show Your Working

Examiners give marks for method, not just the final answer. Always write: the formula, your substitutions, and the simplified result. Even if you make an arithmetic error, you can still earn most of the marks if your method is correct.

9. Practice Problems

Test your understanding with these 15 problems. Try to solve each one on your own before revealing the solution. Remember: practice is the key to mastering probability!

Question 1

A bag contains 4 red, 6 blue, and 5 green marbles. One marble is drawn at random. What is the probability that it is blue?

Total outcomes: 4 + 6 + 5 = 15.

Favorable outcomes (blue): 6.

P(blue) = 6/15 = 2/5 = 0.4 = 40 %

Question 2

Two dice are rolled. What is the probability that the sum is 7?

Total outcomes: 6 × 6 = 36.

Favorable outcomes: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6.

P(sum = 7) = 6/36 = 1/6 ≈ 16.67 %

Question 3

A card is drawn from a standard 52-card deck. What is the probability it is a face card (Jack, Queen, or King)?

Total outcomes: 52.

Favorable: 3 face cards per suit × 4 suits = 12.

P(face card) = 12/52 = 3/13 ≈ 0.2308 ≈ 23.1 %

Question 4

A coin is flipped 3 times. What is the probability of getting exactly 2 Heads?

Total outcomes: 2³ = 8 → {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}.

Favorable (exactly 2 Heads): {HHT, HTH, THH} = 3.

P(exactly 2 Heads) = 3/8 = 0.375 = 37.5 %

Question 5

In a class, P(student wears glasses) = 0.3 and P(student is left-handed) = 0.1. If these are independent, what is the probability a student wears glasses AND is left-handed?

Since independent: P(glasses AND left-handed) = P(glasses) × P(left-handed) = 0.3 × 0.1 = 0.03 = 3 %

Question 6

A bag has 10 balls: 6 red and 4 white. Two are drawn without replacement. What is the probability both are red?

P(1st red) = 6/10. After drawing 1 red: 5 red and 4 white remain (9 total).

P(2nd red | 1st red) = 5/9.

P(both red) = (6/10)(5/9) = 30/90 = 1/3 ≈ 33.3 %

Question 7

What is the probability of drawing a card that is either a Spade or a 10 from a 52-card deck?

P(Spade) = 13/52. P(10) = 4/52. P(Spade AND 10) = 1/52 (10 of Spades).

P(Spade or 10) = 13/52 + 4/52 − 1/52 = 16/52 = 4/13 ≈ 30.77 %

Question 8

The probability of rain on Saturday is 0.4 and on Sunday is 0.3. If these are independent, what is the probability of rain on at least one day?

Complement approach:

P(no rain Sat) = 0.6. P(no rain Sun) = 0.7.

P(no rain either day) = 0.6 × 0.7 = 0.42.

P(rain at least one day) = 1 − 0.42 = 0.58 = 58 %

Question 9

A box has 3 defective and 7 non-defective items. Two items are randomly selected. What is the probability that exactly one is defective?

Case 1: 1st defective, 2nd non-defective = (3/10)(7/9) = 21/90.

Case 2: 1st non-defective, 2nd defective = (7/10)(3/9) = 21/90.

P(exactly 1 defective) = 21/90 + 21/90 = 42/90 = 7/15 ≈ 46.7 %

Question 10

In a survey, 50 % of people like tea, 40 % like coffee, and 20 % like both. What percentage likes neither?

P(T or C) = P(T) + P(C) − P(T and C) = 0.5 + 0.4 − 0.2 = 0.7

P(neither) = 1 − 0.7 = 0.3 = 30 %

Question 11

A password is formed using 4 digits (0–9). What is the probability that all 4 digits are the same?

Total passwords: 10⁴ = 10,000.

Favorable: 0000, 1111, 2222, … , 9999 = 10.

P(all same) = 10/10,000 = 1/1,000 = 0.001 = 0.1 %

Question 12

If P(A) = 0.6, P(B) = 0.5, and P(A | B) = 0.7, find P(A and B).

From conditional probability: P(A | B) = P(A and B) / P(B).

0.7 = P(A and B) / 0.5

P(A and B) = 0.7 × 0.5 = 0.35

Question 13

Five books are placed randomly on a shelf: 2 Math books and 3 Science books. What is the probability that the two Math books are next to each other?

Total arrangements: 5! = 120.

Favorable: Treat the 2 Math books as one block → 4 items → 4! ways = 24. The 2 Math books can be arranged within the block in 2! = 2 ways. Favorable = 24 × 2 = 48.

P(Math books together) = 48/120 = 2/5 = 0.4 = 40 %

Question 14

A test has 5 true/false questions. A student guesses randomly on all questions. What is the probability of getting all 5 correct?

Each question: P(correct) = 1/2. Questions are independent.

P(all 5 correct) = (1/2)⁵ = 1/32 ≈ 0.03125 = 3.125 %

Question 15

Two events A and B are mutually exclusive. P(A) = 0.35, P(B) = 0.25. Find P(A or B) and P(A and B).

Since mutually exclusive: P(A and B) = 0.

P(A or B) = P(A) + P(B) − P(A and B) = 0.35 + 0.25 − 0 = 0.60 = 60 %

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Probability Calculator

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10. Summary of Key Points

Concept / Rule	Formula	When to Use
Basic Probability	P(E) = Favorable / Total	Any simple probability question with equally likely outcomes.
Addition Rule	P(A or B) = P(A) + P(B) − P(A∩B)	Finding the probability of A or B occurring.
Multiplication Rule	P(A and B) = P(A) × P(B\|A)	Finding the probability of A and B both occurring.
Complement Rule	P(A′) = 1 − P(A)	"Not A" problems and "at least one" problems.
Conditional Probability	P(A\|B) = P(A∩B) / P(B)	Finding probability of A given B has occurred.
Independent Events	P(A∩B) = P(A) × P(B)	When events do not affect each other.
Mutually Exclusive	P(A∩B) = 0	Events that cannot happen at the same time.

📌 Final Reminders

Probability values are always between 0 and 1 (or 0 % and 100 %).
The sum of probabilities of all outcomes in a sample space equals 1.
Always define your sample space before solving a problem.
Use tree diagrams and Venn diagrams whenever possible — they make complex problems visual and manageable.
For "at least one" problems, use the Complement Rule — it is almost always the fastest approach.
Show all working on exams — method marks matter!
Practice regularly with timed conditions to build speed and confidence.

🎓 You have completed the Probability Formulas Study Guide!
Bookmark this page for quick revision before your exams. Good luck! 🍀