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Ordinary Level Mathematics Revision Notes

Topic 1.2: Concepts of Probability

These notes explain Concepts of Probability in simple English so students can understand the topic clearly, not just memorise rules. The Ordinary Level version keeps the focus on the core ideas without unnecessary higher-only pressure.

Curriculum Irish Leaving Certificate

Subject Mathematics

Level Ordinary Level

Premium feature Teacher-style explanations and cleaner study flow

Focus

Understanding before memorising

Interactive

Study tools and guided structure

Question Style

Explained examples and exam practice

Format

Website reading and printable notes

Subtopics Covered

The Probability Scale and Basic Probability
Relative Frequency (Experimental Probability)
Mutually Exclusive Events & The Addition Rule (OR)
Independent Events & The Multiplication Rule (AND)
📝 Exam Style Questions
🔥 Challenge Questions (No Peeking!)

What Makes This Version Better

Cleaner Ordinary Level focus with no unnecessary higher-only material
Simple teacher-style explanations before each method
Worked examples in clear stages
Cleaner page flow for student understanding
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Disclaimer This publication is an independent educational resource developed by ExamsLogic and compiled by experienced educators. It is based on publicly available official curricula, including Cambridge, Pearson Edexcel, IB, and the Irish Leaving Certificate. This product is not endorsed by, affiliated with, or sponsored by any examination board or governing authority. All registered trademarks remain the property of their respective owners.

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Learning Objectives

What You Should Be Able To Do

Understand the scale of probability from 0 (impossible) to 1 (certain).
Calculate basic theoretical probability using fractions, decimals, and percentages.
Understand and apply the concept of Relative Frequency (Experimental Probability).
Identify Mutually Exclusive events and use the Addition Rule (OR rule).
Identify Independent events and use the Multiplication Rule (AND rule).

Interactive Study Tools

These tools are here to help students slow down and think about the method before rushing to an answer.

Tool 1

Study Planner

Use this to break a topic into small study sessions.

How many subtopics? How many days?

Plan: A clear breakdown will appear here.

Tool 2

Method Reminder

This quick guide reminds students what to do when they feel stuck.

Read the story: what is the question really asking?
Choose the method: identify the correct idea before calculating.
Show the reason: do not skip the logic.
Check the answer: make sure it fits the situation.

1. The Probability Scale and Basic Probability

Simple explanation: This section is about how likely an event is and how to calculate that likelihood clearly.

Probability is a measure of how likely an event is to happen. It is always a number between 0 and 1. We use the formula:

P(Event) = ^{Number of Favourable Outcomes} / _{Total Number of Possible Outcomes}

Example 1 (Basic): Rolling a fair 6-sided die. What is the probability of rolling a 4?
Favourable outcomes: 1 (just the number 4). Total outcomes: 6. P(4) = 1/6.

Example 2 (Cards): Drawing a card from a standard deck of 52. What is the probability of drawing a King?
There are 4 Kings in a deck. P(King) = 4/52 = 1/13.

Example 3 (The Complement): The probability that an event will NOT happen. If P(Rain) = 0.3, then P(Not Rain) = 1 - 0.3 = 0.7. The sum of all probabilities in a sample space always equals 1.

Examiner Tip

💡 Examiner Tip: Always simplify your fractions! While 4/52 is correct, writing 1/13 guarantees full marks and makes subsequent calculations easier. You can also give answers in decimals or percentages unless fractions are specifically requested.

Examiner Secret

🕵️ Examiner Secret: Standard deck questions are extremely common. Memorize the deck: 52 cards, 4 suits (Hearts, Diamonds, Clubs, Spades), 13 cards per suit, 26 red cards, 26 black cards, 12 picture cards (J, Q, K).

Common Mistake

❌ Common Mistake: Writing a probability as a number greater than 1 (e.g., writing "1.5" or "5"). If your probability calculation gives you a number bigger than 1 or less than 0, you have made an error!

2. Relative Frequency (Experimental Probability)

Simple explanation: This section explains the difference between what should happen in theory and what actually happened in a real experiment.

Theoretical probability tells us what should happen. Relative frequency tells us what actually happened during an experiment.

Relative Frequency = ^{Number of Successful Trials} / _{Total Number of Trials}

Example 1: A drawing pin is dropped 100 times. It lands "point up" 65 times and "point down" 35 times. The relative frequency of landing point up is 65/100 = 0.65.

Example 2: A biased coin is flipped 500 times and gets 320 Heads. Relative frequency of Heads = 320/500 = 0.64. We can use this to estimate that if flipped 1000 times, it would get roughly 640 Heads.

Examiner Tip

💡 Examiner Tip: The Law of Large Numbers states that as the number of trials increases, the relative frequency will get closer and closer to the true theoretical probability. This is a very common 1-mark theory question!

Exam Trap

⚠️ Exam Trap: Do not assume a coin or die is fair unless the question says "fair" or "unbiased." If they give you a table of experimental results, you must use their data (Relative Frequency) rather than the standard 1/6 theoretical probability!

3. Mutually Exclusive Events & The Addition Rule (OR)

Simple explanation: This section helps students understand when probabilities are added, and why overlap matters.

Events are Mutually Exclusive if they cannot happen at the same time. For example, drawing a card that is both a Heart and a Spade is impossible. When asked for the probability of Event A OR Event B, we ADD.

If Mutually Exclusive: P(A or B) = P(A) + P(B)

If NOT Mutually Exclusive: P(A or B) = P(A) + P(B) - P(A and B)

Example 1 (Mutually Exclusive): Rolling a die. What is the probability of rolling a 2 OR a 5?
P(2) = 1/6. P(5) = 1/6. P(2 or 5) = 1/6 + 1/6 = 2/6 = 1/3.

Example 2 (Not Mutually Exclusive): Drawing a card. Probability of a King OR a Heart?
P(King) = 4/52. P(Heart) = 13/52. But wait! The King of Hearts was counted twice! We must subtract the overlap.
P(King or Heart) = 4/52 + 13/52 - 1/52 = 16/52 = 4/13.

Examiner Secret

🕵️ Examiner Secret: The word "OR" in probability almost always means you need to use a PLUS sign.

Common Mistake

❌ Common Mistake: Forgetting to subtract the intersection (the overlap) when events are not mutually exclusive. Always ask yourself: "Can these two things happen to the same object at the same time?"

4. Independent Events & The Multiplication Rule (AND)

Simple explanation: This section explains when probabilities are multiplied and how one event can affect the next.

Events are Independent if the outcome of one does not affect the outcome of the other (e.g., flipping a coin and then rolling a die). When asked for the probability of Event A AND Event B, we MULTIPLY.

If Independent: P(A and B) = P(A) × P(B)

Example 1 (Independent): Flipping a coin twice. Probability of getting Heads AND Heads?
P(H) = 1/2. P(H) = 1/2. P(H and H) = 1/2 × 1/2 = 1/4.

Example 2 (With Replacement): A bag has 3 Red and 2 Blue marbles. You draw a marble, put it back, and draw another. P(Red and Red)?
P(R) = 3/5. P(R) = 3/5. P(R,R) = 3/5 × 3/5 = 9/25.

Example 3 (Without Replacement - Dependent!): A bag has 3 Red and 2 Blue marbles. You draw a marble, keep it out, and draw another. P(Red and Red)?
First draw P(R) = 3/5. Second draw: There are now only 2 Red left, and only 4 total marbles left. P(R) = 2/4. Total = 3/5 × 2/4 = 6/20 = 3/10.

Examiner Tip

💡 Examiner Tip: The word "AND" usually means MULTIPLY.

Exam Trap

⚠️ Exam Trap: Watch out for the phrases "with replacement" vs "without replacement". If an item is NOT replaced, the total number on the bottom of the fraction for the next draw MUST decrease by 1!

📝 Exam Style Questions

Simple explanation: This section is about how likely an event is and how to calculate that likelihood clearly.

Easy Q1: A spinner has 8 equal sections numbered 1 to 8. What is the probability of spinning an even number or a 7?

Medium Q2: A bag contains 5 green, 4 yellow, and 3 black counters. Two counters are drawn at random without replacement. Find the probability that both counters are yellow.

Hard Q3: Event A and Event B are independent. P(A) = 0.6 and P(B) = 0.3. Find P(A or B).

A1: Even numbers = {2, 4, 6, 8} (4 outcomes). Number 7 = 1 outcome. These are mutually exclusive. P(Even) + P(7) = 4/8 + 1/8 = 5/8.

A2: Total counters = 12. P(First is Yellow) = 4/12. Since it is without replacement, there are now 3 yellow left out of 11 total. P(Second is Yellow) = 3/11. P(Y and Y) = 4/12 × 3/11 = 12/132 = 1/11.

A3: Because they are not mutually exclusive, P(A or B) = P(A) + P(B) - P(A and B).
Since they are independent, we find the overlap by multiplying: P(A and B) = 0.6 × 0.3 = 0.18.
P(A or B) = 0.6 + 0.3 - 0.18 = 0.9 - 0.18 = 0.72.

🔥 Challenge Questions (No Peeking!)

Simple explanation: This section is about how likely an event is and how to calculate that likelihood clearly.

Easy C1: In a class of 30 students, 18 play soccer, 15 play basketball, and 5 play neither. If a student is chosen at random, what is the probability they play both sports?

Medium C2: A fair coin is flipped 4 times. What is the probability of getting exactly one Head?

Hard (HL) C3: You have two identical boxes. Box A contains 3 gold coins and 2 silver coins. Box B contains 1 gold coin and 4 silver coins. You randomly pick a box, and then randomly draw a coin from it. What is the probability that the coin you draw is gold?

(Answers at the very bottom of the page)

📌 Quick Summary

1. Basic Probability: Favourable / Total. Must be between 0 and 1.
2. Relative Frequency: Based on real experiments. Gets closer to theoretical probability with more trials.
3. OR means ADD: Watch out for overlaps (subtract P(A and B) if they are not mutually exclusive).
4. AND means MULTIPLY: Watch out for "without replacement" changing the total count.

🎓 Final Examiner Advice

Whenever you see a probability question, underline key words like OR, AND, REPLACEMENT, and FAIR. Drawing a quick visual aid, like a tree diagram for successive events or a Venn diagram for overlapping sets, often makes the calculation completely obvious. Don't try to do it all in your head!

Challenge Answers:
C1: Draw a Venn diagram. Total = 30. Subtract the 5 who play neither = 25 playing sports. Soccer (18) + BBall (15) = 33. Since there are only 25 players, the overlap (Both) must be 33 - 25 = 8. Probability = 8/30 = 4/15.
C2: The sequences with exactly one head are HTTT, THTT, TTHT, TTTH (4 paths on a tree diagram). Each specific path has a probability of 1/2 * 1/2 * 1/2 * 1/2 = 1/16. So, 4 * 1/16 = 4/16 = 1/4.
C3: Use a tree diagram. P(Box A) = 1/2, P(Gold from A) = 3/5. (Path A->Gold = 1/2 * 3/5 = 3/10). P(Box B) = 1/2, P(Gold from B) = 1/5. (Path B->Gold = 1/2 * 1/5 = 1/10). Total P(Gold) = 3/10 + 1/10 = 4/10 = 2/5.