How to Master Probability & Tree Diagrams in ECZ Mathematics

Mathematics (Grade 12 & GCE)  |  Updated March 2026  |  12 min read

In the ECZ Grade 12 and GCE Mathematics Paper 2 examinations, Probability and Tree Diagrams appear as a predictable, high-scoring section worth 6 to 8 marks. Mastering tree diagrams requires clear rules on independent vs. dependent events, fraction multiplication, and branch addition.

For Senior Secondary Candidates

Tree diagram questions in ECZ exams usually follow a standard pattern: drawing the incomplete tree diagram, completing missing branch probabilities, and calculating combined probabilities for events (with or without replacement).

1. Essential Fundamentals of Probability

The probability \(P(E)\) of an event occurring is defined as:

\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]

Key probability rules to remember:

  • \(0 \le P(E) \le 1\): Probability is never negative and never greater than 1.
  • Sum of probabilities of all possible mutually exclusive outcomes equals 1: \(P(E) + P(\text{not } E) = 1\).

2. Dependent vs. Independent Events

Understanding the distinction between selection with replacement and without replacement is critical:

  • With Replacement (Independent Events): The chosen item is put back. Total count stays constant for the second pick.
  • Without Replacement (Dependent Events): The chosen item is NOT put back. Total items and the specific color count decrease by 1 for the second pick.
Exam Alert: Watch the Keyword!

Always highlight phrases in the question such as "picked at random one after another WITHOUT replacement" or "replaced before the second pick". This single phrase dictates your entire branch fraction logic!

3. Step-by-Step Worked Example (ECZ Past Paper Style)

Problem Statement:

A bag contains 5 red pens and 3 blue pens. Two pens are picked at random from the bag, one after the other, without replacement.

  1. Draw a complete tree diagram showing all probabilities.
  2. Find the probability that both pens picked are red: \(P(\text{Red, Red})\).
  3. Find the probability that the pens are of different colors: \(P(\text{Different Colors})\).

Step 1: First Selection Branch (Total = 5 + 3 = 8 pens)

  • \(P(\text{Red}_1) = \frac{5}{8}\)
  • \(P(\text{Blue}_1) = \frac{3}{8}\)

Step 2: Second Selection Branch (Total remaining = 7 pens)

If first was Red (4 red remaining out of 7):

  • \(P(\text{Red}_2 \mid \text{Red}_1) = \frac{4}{7}\)
  • \(P(\text{Blue}_2 \mid \text{Red}_1) = \frac{3}{7}\)

If first was Blue (2 blue remaining out of 7):

  • \(P(\text{Red}_2 \mid \text{Blue}_1) = \frac{5}{7}\)
  • \(P(\text{Blue}_2 \mid \text{Blue}_1) = \frac{2}{7}\)

Step 3: Calculating Probabilities

Part (ii) - Both Red:

\[ P(\text{Red, Red}) = \frac{5}{8} \times \frac{4}{7} = \frac{20}{56} = \frac{5}{14} \]

Part (iii) - Different Colors (Red then Blue OR Blue then Red):

\[ P(\text{Red, Blue}) = \frac{5}{8} \times \frac{3}{7} = \frac{15}{56} \] \[ P(\text{Blue, Red}) = \frac{3}{8} \times \frac{5}{7} = \frac{15}{56} \] \[ P(\text{Different Colors}) = \frac{15}{56} + \frac{15}{56} = \frac{30}{56} = \frac{15}{28} \]

4. The Golden Rules of Tree Diagrams

  1. Multiply Along Branches: To find the probability of a sequence of events along a path, multiply the fractions along that path.
  2. Add Across Branches: To find the total probability of multiple favorable outcome paths, add the results of each path together.
  3. Check Branch Totals: At any decision node, the fractions on the branching arms must sum to 1 (e.g., \(\frac{5}{8} + \frac{3}{8} = 1\)).
Teacher & Tutor Tip

Remind learners to leave fractions in unsimplified form (e.g., \(\frac{20}{56}\)) until final addition steps are complete. This keeps common denominators matching and avoids fraction arithmetic errors!

Frequently Asked Questions

Q1: What is the difference between replacement and non-replacement in probability?

With replacement, an item chosen is returned to the container, so total outcomes and probabilities remain identical for the next pick. Without replacement (non-replacement), the item is NOT returned, so both the total items and the specific item count decrease by 1 for the second pick.

Q2: When do you multiply vs add probabilities on a tree diagram?

Multiply probabilities ALONG a branch (e.g., P(Red then Blue) = P(Red) * P(Blue)). Add probabilities ACROSS different branches that satisfy the condition (e.g., P(one Red and one Blue) = P(Red, Blue) + P(Blue, Red)).

Q3: Are probability tree diagrams tested in ECZ Maths Paper 1 or Paper 2?

Probability tree diagrams are consistently tested in ECZ Senior Secondary Mathematics Paper 2 (Section B), carrying 6 to 8 marks.

Ace Your ECZ Mathematics Exams

Practice past ECZ Grade 12 & GCE Paper 2 probability questions with full step-by-step solutions.

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