Probability

2026 What You Need to Know (Syllabus Objectives)

By the end of this topic, you should be able to:

  1. Calculate probabilities in simple situations by listing all equally-likely outcomes, or by using permutations and combinations (for example: rolling dice, drawing coloured balls from a bag).

  2. Use addition and multiplication rules for probabilities when appropriate.

  3. Understand exclusive events and independent events, including how to test whether two events are independent by checking if P(A and B) = P(A) × P(B).

  4. Calculate and use conditional probabilities in simple cases using sample spaces, tree diagrams, or the formula P(A|B) = P(A and B) ÷ P(B).

1. Basic Probability Concepts

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What is Probability?

Probability is a number between 0 and 1 that tells us how likely something is to happen.

  • Probability = 0 means the event is impossible (it will never happen).
  • Probability = 1 means the event is certain (it will definitely happen).
  • Probability between 0 and 1 means the event might happen.

How to Calculate Probability

The basic formula for probability is:

P(Event) = Number of favourable outcomes ÷ Total number of possible outcomes

This only works when all outcomes are equally likely (each outcome has the same chance of happening).

2. Evaluating Probabilities by Enumeration

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Enumeration means listing all possible outcomes to find a probability.

Method: Enumeration of Equiprobable Events

Equiprobable means "equally likely" – every outcome has the same chance.

Steps:

  1. List all possible outcomes (this is called the sample space).
  2. Count how many outcomes are favourable (match what you want).
  3. Count the total number of outcomes.
  4. Use the formula: P(Event) = Favourable outcomes ÷ Total outcomes.

Example 1: Rolling Two Fair Dice

Problem: Two fair six-sided dice are rolled. What is the probability that the total score is 7?

Solution:

Step 1: List all possible outcomes.

  • Each die shows 1, 2, 3, 4, 5, or 6.
  • Total possible outcomes = 6 × 6 = 36 (we can represent these as pairs like (1,1), (1,2), ..., (6,6)).

Step 2: Find favourable outcomes (pairs that sum to 7):

  • (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
  • There are 6 favourable outcomes.

Step 3: Calculate probability:

  • P(total = 7) = 6/36 = 1/6

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