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Probability: replacement, tree diagrams and checking outcomes

Probability starts with a clearly defined experiment. Decide whether outcomes are equally likely and whether a selection changes what is available for the next selection. The words ‘without replacement’ change the second denominator.

Choose the method

For equally likely outcomes, probability = favourable outcomes/total outcomes. The answer must lie between 0 and 1.

Multiply conditional probabilities along a tree branch. Add the probabilities of mutually exclusive branches that produce the event you want.

Use a complement for ‘at least one’ when it is easier: P(at least one) = 1 - P(none). Independence must be given or justified; separate events are not automatically independent.

Worked examples

Example 1

A bag contains 3 red and 2 blue counters. Two are drawn without replacement. Find P(both red).

The first draw is red with probability 3/5.
After a red draw, 2 red counters remain among 4 counters. The second probability is 2/4.
P(both red) = 3/5 × 2/4 = 3/10.

Example 2

For the same bag, find P(one red and one blue) without replacement.

Red then blue: 3/5 × 2/4 = 3/10.
Blue then red: 2/5 × 3/4 = 3/10.
These branches cannot happen together, so add: 3/10 + 3/10 = 3/5.

Try it yourself

A fair die is rolled twice independently. Find P(at least one six).

Show the worked answer

The complement is no six on either roll: (5/6)² = 25/36.
Required probability = 1 - 25/36 = 11/36.

Common mistakes

Count both orders when the event can occur in either order.
With replacement, the original probabilities return. Without replacement, update the counts after every draw.

Sources and review

SEAB 2026 syllabus: O-Level Mathematics 4052: probability.

Original Rae practice, prepared with AI assistance. Selected numerical results and their displayed working are automatically checked at publication; this does not verify every explanation. Curriculum references checked on 5 September 2026. No teacher review or SEAB endorsement is claimed.

What the syllabus expects

probability as a way of measuring chance
the probability of single events
Scope: includes listing every possible outcome in a simple chance setting to compute the probability
the probability of simple combined events
Scope: includes using possibility diagrams and tree diagrams where suitable
adding and multiplying probabilities for mutually exclusive events and for independent events

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More O-Level Elementary Maths (E-Maths) topics

Whole numbers and the operations performed on them · Ratio and proportion · Percentage · Rate and speed · Algebraic expressions and formulae · Functions together with their graphs · all of O-Level Elementary Maths (E-Maths)