11+ Probability: Equally Likely Outcomes and Worked Answers

A bag containing red and blue counters does not automatically give each colour a half chance. If there are more red counters than blue, a random draw is more likely to select red. Counting the colour names misses the individual objects that can actually be selected.

These original explanations and questions support preparation for 11-plus exams taken in 2027. They are not official papers or a complete school syllabus. Check your target school’s guidance and try each question before opening the explanation. Written English models show a supported response, not the only acceptable wording.

Define the experiment before calculating

For a finite set of equally likely outcomes, probability is the number of favourable outcomes divided by the total number of outcomes. Say what is random and why the outcomes can be treated equally. A fair six-sided die has six equally likely numbered faces; a spinner needs equal sectors and a fair mechanism before equal-sector counting is justified.

Values between zero and one-half are less likely than not; values between one-half and one are more likely than not. These descriptions do not promise what will happen on a particular attempt. Even a high-probability outcome can fail to occur.

Worked example: count objects, not categories

A bag has five red counters and three blue counters, all equally likely to be drawn. The chance of red is 5/8, not 1/2. The chance of not red is 3/8. Together those complementary events cover every counter once, so their probabilities total one. This complement method works when “not” means every outcome outside the named event.

Worked example: why replacement matters

Start with three green counters and two yellow counters. Draw one uniformly at random, replace it, mix, then draw again. The probability of two greens is 3/5 × 3/5 = 9/25. Without replacement, after a green has been removed there are two greens among four counters: 3/5 × 2/4 = 3/10. Updating the second fraction is essential; the first draw changes the bag.

For two fair independent coin tosses, write the ordered outcomes HH, HT, TH and TT. “One head and one tail” covers HT and TH, not just one outcome. A list is often clearer than a shortcut, particularly when order matters or several routes lead to the same event.

Eight original questions

Check the model as well as the arithmetic

An answer greater than one or below zero cannot be a probability. A denominator based on colour names rather than counters needs correcting. A repeated-draw calculation needs an explicit replacement rule. None of these theoretical models predicts an actual school result or pass rate.

The Stanford probability teaching notes explain equally likely sample spaces, complements and replacement. This guide is an introduction to selected reasoning problems, not a claim that probability appears in every 11-plus specification.

Choose a useful next step

Use fraction and decimal conversions to express the same probability differently, and data interpretation to distinguish observed counts from a model of chance. Return to the maths learning hub or explore original practice and worked answers.

Sources and scope

The DfE programme of study provides a curriculum reference, not a promise about every school’s test. Examples and questions on this page are original; no school affiliation or educator endorsement is implied.