Conditional Probability

2026 Syllabus Objectives

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

  1. Calculate conditional probability using Venn diagrams, tree diagrams and tables.
  2. Understand that you do not need to know special notation or formulas for this topic.

What is Conditional Probability?

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Conditional probability is the probability of something happening when you already know that something else has happened.

The key phrase to look out for is "given that". When you see this phrase, you are being asked to find a conditional probability.

The Basic Idea

Imagine you have a bag of coloured counters. If you pick a counter at random, you would normally calculate the probability out of all the counters in the bag. But what if someone tells you "the counter you picked is red"? Now you only need to think about the red counters, not all the counters. This is conditional probability - you are working with a smaller, restricted set of outcomes.

Example: A computer randomly selects a digit from 1, 2, 3, 4, 5, 6, 7, 8, 9.

  • The probability it selects a multiple of three = 3/9 (there are 3 multiples of three: 3, 6, 9)
  • This is a normal probability (not conditional)

But if the computer is programmed to only select from even numbers, then:

  • The probability it selects a multiple of three given that it selects an even number = 1/4
  • Why? Because there is only 1 multiple of three among the even numbers (which are 2, 4, 6, 8)
  • You calculate this out of 4 possibilities, not 9
  • This is a conditional probability

Calculating Conditional Probability Using Venn Diagrams

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A Venn diagram shows sets of numbers or objects using circles. When calculating conditional probability with a Venn diagram, you need to focus only on the set that has "already happened".

How to do it:

Step 1: Identify which set has "already happened" - this is mentioned after the words "given that".

Step 2: Count how many items are in that set. This becomes your denominator (bottom number of your fraction).

Step 3: Count how many items satisfy both conditions (the original event AND the given condition). This becomes your numerator (top number of your fraction).

Step 4: Write your answer as a fraction.

Example:

A Venn diagram shows:

  • Set A contains: 2, 6, 12, 14, 28
  • Set B contains: 7, 14, 21, 28, 35
  • The numbers 14 and 28 are in both sets

Question: A number is picked at random. Find the probability that the number is in set A, given that it is in set B.

Solution:

  • "Given that it is in set B" means we only look at set B
  • Set B has 5 numbers: 7, 14, 21, 28, 35
  • Out of these 5 numbers, 2 are also in set A: 14 and 28
  • Probability = 2/5

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