Introduction to Visualising Probability

Probability is often about counting how many ways something can happen. When problems get complicated—like rolling two dice or picking two socks from a drawer—it is very easy to miss an outcome. In this chapter, we learn how to use Sample Spaces, Venn Diagrams, and Tree Diagrams to organise our thoughts and ensure we never miss a single possibility!

1. Sample Space Diagrams

A Sample Space is simply a list of all possible outcomes of an experiment. When we have two events happening (like rolling two dice), a Grid is the best way to see every outcome clearly.

How to Draw a Sample Space Grid

Imagine you roll a fair six-sided die and flip a coin. We can list the outcomes in a table:

Die Outcomes: \(1, 2, 3, 4, 5, 6\)
Coin Outcomes: \(H, T\)

The grid would look like this:

\( (H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6) \)
\( (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6) \)

There are \(2 \times 6 = 12\) possible outcomes in total. If you want to find the probability of getting a Head and an even number, you just count them: \( (H, 2), (H, 4), (H, 6) \). That is \(3\) out of \(12\), so \(P(\text{Head and Even}) = \frac{3}{12} = \frac{1}{4}\).

Quick Tip: The Sum of Two Dice

In exams, you are often asked about the sum of two dice. Always draw a \(6 \times 6\) grid and add the numbers together. You will notice a pattern: the most likely sum is \(7\)!

Key Takeaway: Use a grid whenever you have two independent events with a small, fixed number of outcomes.

2. Venn Diagrams

Venn Diagrams use overlapping circles to show the relationship between different sets of data. They are perfect for "survey" style questions (e.g., "30 students like Math, 20 like English...").

The Parts of a Venn Diagram

  • The Rectangle \((\xi)\): This represents the "Universal Set"—everyone or everything being considered.
  • The Circles: These represent specific events or groups (e.g., Set \(A\) and Set \(B\)).
  • The Intersection (\(A \cap B\)): The overlap in the middle. These are things that belong to both \(A\) and \(B\).
  • The Union (\(A \cup B\)): Everything inside both circles. These are things that belong to \(A\) or \(B\) (or both).

The Addition Rule

On your Edexcel formula sheet, you will see:
\(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\)

This is because if you add the whole of circle \(A\) and the whole of circle \(B\), you have counted the middle overlap twice! Subtracting it once makes the answer correct.

Example:

In a group of 10 students, 7 like Pizza (\(P\)), 5 like Burgers (\(B\)), and 3 like both.
1. Start with the middle: Put \(3\) in the overlap.
2. Pizza only: \(7 - 3 = 4\).
3. Burgers only: \(5 - 3 = 2\).
4. Neither: Add them up (\(4 + 3 + 2 = 9\)). Since there are 10 students, \(10 - 9 = 1\) person goes outside the circles.

Key Takeaway: Always fill in the "Both" (intersection) section first!

3. Tree Diagrams

Tree diagrams are the best tool for "multi-stage" events (e.g., "picking a marble, then picking another"). They show the "branches" of what could happen at each step.

The Golden Rules of Tree Diagrams

  1. Multiply across: To find the probability of two things happening one after the other, multiply the probabilities along the branches.
  2. Add down: If there are multiple "winning" paths at the end, add their total probabilities together.
  3. The "1" Rule: The branches coming from a single point must always add up to \(1\).

Independent vs. Dependent (H)

Independent: The probability stays the same. (e.g., flipping a coin twice).
Dependent (Conditional): The probability changes because something was not replaced. (e.g., eating a chocolate from a box makes there be one fewer chocolate left). (Higher tier students must pay close attention to "without replacement" wording).

Example:

The probability of it raining is \(0.3\). If it rains, the probability I am late is \(0.8\). If it is dry, the probability I am late is \(0.1\).
- Path 1 (Rain and Late): \(0.3 \times 0.8 = 0.24\)
- Path 2 (Dry and Late): \(0.7 \times 0.1 = 0.07\)
- Total probability of being late: \(0.24 + 0.07 = 0.31\)

Key Takeaway: "AND" means multiply (\(\times\)), "OR" means add (\(+\)).

4. Conditional Probability (Higher Tier Only)

Conditional probability is the probability of an event \(A\) occurring, given that event \(B\) has already happened. We write this as \(P(A|B)\).

Using Diagrams for Conditional Probability

  • Venn Diagrams: If the question says "Given that the student likes Pizza...", your "world" shrinks to only the Pizza circle. The denominator of your fraction will be the total number of people in the Pizza circle, not the whole class.
  • Tree Diagrams: The second set of branches will have different denominators. If there are 10 beads and you take one out, the next set of branches must be out of \(9\).

The Formula: \(P(A \text{ and } B) = P(A|B) \times P(B)\). This is just a fancy way of saying "multiply along the branches"!

Common Mistakes to Avoid

  • Forgetting the "Outside": In Venn diagrams, always check if there are items that don't fit into either circle. They must stay inside the rectangle!
  • Not Simplifying: Always check if your final probability fraction can be simplified, though in many GCSE papers, unsimplified fractions are accepted unless "simplest form" is requested.
  • Replacement: Read carefully! Does the person put the marble back in the bag? If not, the denominator must go down by \(1\) for the second branch.

Quick Review:
- Grids: Great for two dice or simple combinations.
- Venn: Great for "some," "both," and "none" survey data.
- Tree: Great for "first this, then that" sequences.