Probability: apply the property that the probabilities of an exhaustive set of outcomes sum to one; apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to one

Overview

Probability is a number that describes how likely an outcome is: it runs from 0 for impossible to 1 for certain. In Pearson Edexcel GCSE Mathematics 1MA1 P4, you use the fact that the probabilities of all separate, exhaustive elementary outcomes add to 1. That one fact lets you find a missing decimal, fraction or algebraic expression, and it also supports complements such as ‘not red’. P4 is assessed at both Foundation and Higher; linear algebraic missing-probability tables are core for both tiers, so a challenging question is not automatically Higher-only 1 2.

The key is to name precisely what is being added. Individual elementary outcomes in a sample space cannot happen together, so their probabilities can be added. Larger events can overlap, however, and then adding them double-counts shared outcomes. This guide develops the method, explains why it works, and gives original, examiner-style practice rather than Pearson questions. Pearson qualification papers 1, 2 and 3 are each 90 minutes and 80 marks: Paper 1 is non-calculator, while Papers 2 and 3 allow a calculator 1.

Key Concepts

1. Exhaustive outcomes: nothing is missing

An outcome is one possible individual result, such as rolling a 4. A sample space contains every possible outcome under the conditions of the experiment. For a standard six-sided die, the sample space is {1, 2, 3, 4, 5, 6}.

A collection is exhaustive when it covers every possibility. If a bag contains only red, blue and green counters, those three colours are exhaustive. If yellow counters are also present, a list containing only red, blue and green is not exhaustive.

This distinction matters before any calculation. Subtracting the listed probabilities from 1 finds the probability of everything omitted together. It identifies one particular missing colour only when that colour is the only remaining possibility.

Cover and recall: A bag contains red, blue, green and yellow counters. You know the probabilities of red and blue. What does the remainder represent? It represents green or yellow together, not green alone.

2. Mutually exclusive events: no double-counting

Events are mutually exclusive when they cannot occur together on one trial. Taking one red counter and taking one blue counter are mutually exclusive. Taking a red counter and taking a large counter need not be: a counter could be both red and large.

If the categories are both exhaustive and mutually exclusive, exactly one category occurs. Their probabilities therefore add to 1. Remember the complete condition: every possibility, counted once.

Mutually exclusive events do not need to be exhaustive for their probabilities to be added. Exhaustiveness is the extra condition that makes the complete sum equal to 1.

Do not confuse mutually exclusive with independent. Independent means one event occurring does not change the probability of another. Mutually exclusive means they cannot happen together. Two events with positive probabilities that are mutually exclusive are not independent: once one occurs, the other becomes impossible.

3. Missing probabilities: find the remainder

For separate, exhaustive categories, add the known probabilities and subtract their total from 1.

Example: A bag contains only red, blue and green counters. P(red) = 0.20 and P(blue) = 0.45.

Known total = 0.20 + 0.45 = 0.65.

P(green) = 1 − 0.65 = 0.35.

Check: 0.20 + 0.45 + 0.35 = 1.

The subtraction works because green occupies all the probability left after the known colours. Candidates should display the calculation rather than write an unexplained answer. Under a question’s particular mark scheme, correct working may earn method credit even if subsequent arithmetic is incorrect; there is no universal mark awarded simply for writing a sum 4.

For fractions, make the whole compatible with the denominator. If known probabilities are 2/7 and 3/7, the missing probability is 7/7 − 5/7 = 2/7. With percentages, subtract from 100%, not from 1.

4. Equal missing entries: share the remainder

Two blanks do not automatically have equal probabilities. Equality must be stated or established from the information.

Pearson teaching guidance includes an authentic specimen question with red 0.24, yellow 0.32, and equal probabilities for blue and green. The known total is 0.56, leaving 0.44. Blue and green share this equally, so each probability is 0.22. That specimen question was worth 2 marks in total 2. Its official scheme governs the precise credit for a candidate’s response.

A common error is giving each missing colour the whole remainder. That allocates the same probability twice. If three equal entries were missing, the remainder would instead be divided by 3.

Why does this work? The remainder belongs to the missing entries collectively. Only the stated relationship tells you how to distribute it.

5. Complements: the probability of ‘not’

The complement of event A, written A′ or ‘not A’, contains every outcome in the sample space outside A. Either A happens or it does not. These two events are mutually exclusive and exhaustive, so P(A′) = 1 − P(A).

In the three-colour bag, P(red) = 0.20. Therefore P(not red) = 0.80. Not red includes blue and green: 0.45 + 0.35 = 0.80. It does not mean blue alone.

The complement rule is often quicker than adding several other categories. If P(rain) = 0.35, then P(not rain) = 0.65, provided the event refers to the same defined period and conditions.

Cover and recall: Explain why an event and its complement always total 1. Your answer should mention complete coverage and no overlap.

6. Algebraic probability tables

When entries contain a letter, form an equation by setting the total equal to 1. Collect like terms, solve, and substitute the value back into every expression.

For probabilities x, 2x, 0.15 and 0.25:

x + 2x + 0.15 + 0.25 = 1.

3x + 0.40 = 1, so 3x = 0.60 and x = 0.20.

The completed probabilities are 0.20, 0.40, 0.15 and 0.25. They total 1 and all lie within the allowed range.

Do not stop at x if the question asks for an actual probability. An entry such as 2x must be evaluated. A question asking for ‘not B’ then requires the complement of P(B), not the complement of x.

Linear algebraic missing-probability tables belong to both tiers 2. Higher-only extension: conditional probability, P9, is a related but distinct topic and is not required for the P4 practice here 1.

7. Overlap and invalid distributions

On a fair die, ‘even’ is {2, 4, 6} and ‘above 3’ is {4, 5, 6}. They overlap at 4 and 6. Their union contains {2, 4, 5, 6}, so its probability is 4/6 = 2/3, not 1.

An exhaustive collection of overlapping events need not have probabilities that sum to 1. Exhaustiveness alone is insufficient for the simple addition rule.

A total of 1 is also insufficient to prove a table is valid. The entries 0.9, −0.1 and 0.2 total 1, but −0.1 cannot be a probability. Check both the total and every individual value.

Mathematical Relationships

Here P(A) means the probability of event A, A′ means its complement, and ‘A or B’ includes either event or both.

The general addition formula is printed on the 2026 Foundation sheet. The exhaustive-total and complement rules are not separately printed there. Learn these rules rather than assuming that every probability relationship will be supplied. Formula-sheet statements here are specific to the checked 2026 sheet, not promises about later examination series 3.

Must memorise conversions: decimal p corresponds to 100p%; q% corresponds to q/100. Thus 0.35 = 35% = 7/20, not 0.35%. Separate exhaustive percentages total 100%. Probabilities have no physical units, so do not append cm, kg or seconds.

Practical Applications and Data Skills

Probability tables model spinner colours, game prizes or categories such as pass, rework and reject. First check whether every item receives exactly one category. If an item can be both ‘needs rework’ and ‘rejected’, the categories overlap and cannot be treated as separate parts of a total without adjustment.

Read probability scales from 0 to 1 unless percentages are explicitly used. For probability bar charts, match each bar to its category and read its height against the vertical scale. Separate exhaustive categories have probabilities totalling 1. Distinguish probability heights from frequency counts, which total the number of observations rather than 1.

There is no required practical for this GCSE Mathematics topic 1. Optional repeated counter draws or spinner trials illustrate relative frequency. Experimental proportions need not match theoretical probabilities exactly, especially with few trials. Use observed data when asked to estimate; use the stated model for theoretical calculations.

Exam Technique

Underline the requested event and read every named outcome. ‘Only’ can establish complete coverage; ‘equal’ tells you to share a remainder; ‘not’ signals a complement. Check overlap before adding events.

Pearson’s papers each allow 90 minutes for 80 marks, giving an overall planning average of 67.5 seconds per mark 1. This is not a rigid deadline for each question. Complete direct one-mark parts briskly and preserve time for linked algebra and checking.

For a one-mark ‘give’ part, state the requested value. For a two-mark calculation, show the subtraction and final answer. For longer questions, display the total equation, solution, substitution and requested probability. These are working strategies, not universal official allocations.

Pearson marking guidance generally accepts probability answers as fractions, decimals or percentages 4. Match any explicitly requested form and preserve exact fractions where useful. Odds notation such as 2:3 is not a probability answer. Avoid rounding partway through an algebraic calculation.

Cover and Recall: Audio Revision

Listen to the ten-minute episode: introduction 0:00–1:00, core concepts 1:00–6:00, exam tips 6:00–8:00, recall quiz 8:00–9:00, and recap 9:00–10:00. Pause when prompted and calculate before hearing the answer.

After listening, close the guide and recreate the red, blue and green example. Explain why the colours total 1, then calculate not red in two ways. Tomorrow, solve an algebraic table without looking. Return to it a few days later: retrieval practice is more demanding, and more useful, than simply recognising a familiar worked solution.

References

The worked examples and practice questions are original teaching questions with indicative schemes, not official Pearson questions.