Probability of Combined Events

2026 Syllabus Objectives

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

  1. Calculate the probability of combined events using sample space diagrams, Venn diagrams, and tree diagrams
  2. Work with combined events both with replacement and without replacement
  3. Use the notation P(A ∩ B) and P(A ∪ B) in the context of Venn diagrams
  4. Draw tree diagrams correctly with outcomes at the end of branches and probabilities by the side of branches

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What Are Combined Events?

Combined events happen when you have two or more things happening together or one after another. For example, flipping a coin twice, rolling two dice, or picking two cards from a deck.

To work out the probability of combined events, we use three main tools:

  • Sample space diagrams (grids that show all possible outcomes)
  • Venn diagrams (circles that show overlapping groups)
  • Tree diagrams (branching diagrams that show sequences of events)

Sample Space Diagrams

A sample space diagram is a grid or table that shows all the possible outcomes when two events happen together.

Example: Rolling Two Dice

When you roll two dice, each die can land on 1, 2, 3, 4, 5, or 6. If you roll a red die and a blue die together, there are 6 × 6 = 36 possible outcomes in total.

You can draw a grid where:

  • The horizontal axis shows the outcomes for the red die (1 to 6)
  • The vertical axis shows the outcomes for the blue die (1 to 6)
  • Each point on the grid represents one possible outcome, like (3, 5) meaning red die shows 3 and blue die shows 5

How to find probabilities using a sample space diagram:

  1. Count how many outcomes give you the result you want
  2. Divide by the total number of possible outcomes (36 for two dice)

Example 1: Find the probability of rolling the same number on both dice.

The outcomes where both dice match are: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6)

There are 6 matching outcomes out of 36 total.

P(both same) = 6/36 = 1/6

Example 2: Find the probability that the sum of the two dice is 8.

The outcomes that add up to 8 are: (2,6), (3,5), (4,4), (5,3), (6,2)

There are 5 outcomes that work.

P(sum is 8) = 5/36

Example 3: Find the probability that when you multiply the two numbers, you get 12.

The outcomes where the product is 12 are: (2,6), (3,4), (4,3), (6,2)

There are 4 outcomes.

P(product is 12) = 4/36 = 1/9

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