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A sample space diagram is a two-way grid listing every outcome of two combined events, with one event across the top and the other down the side. Free notes.
Systematic Enumeration Using Tables, Grids and Venn Diagrams is part of Sample Spaces and Systematic Enumeration in GCSE AQA Higher Maths.
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GCSE AQA Higher
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A sample space diagram is a two-way grid listing every outcome of two combined events, with one event across the top and the other down the side. Free notes.
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Use this lesson to stay inside Sample Spaces and Systematic Enumeration while connecting the detail back to the wider Probability topic.
Read the core explanations from Systematic Enumeration Using Tables, Grids and Venn Diagrams before testing yourself from memory.
Imagine rolling two dice and trying to find the probability of getting a total of 7. If you list outcomes randomly, you will almost certainly miss some — and a missed outcome means a wrong probability. Systematic enumeration is the discipline of listing every possible outcome in a structured way so that nothing is overlooked.
the examiner expects you to use three specific tools for this: tables, grids, and Venn diagrams. Each tool suits a different type of problem, and knowing which to reach for is half the battle. This lesson builds all three skills from the ground up.
When two independent events each have a small number of outcomes, a sample space diagram (also called a possibility space or grid) is the most reliable enumeration tool. You place the outcomes of one event along the top and the outcomes of the other down the side, then fill every cell with the combined result.
The power of a grid is that its structure forces you to account for every combination. You cannot accidentally skip a row or column the way you might when writing a list. Every cell represents exactly one outcome, so counting is straightforward.
Worked example: Building a Sample Space Grid: Two Dice
Two fair six-sided dice are rolled. Find the probability that the total is 7.
Answer: $P(\text{total} = 7) = \dfrac{1}{6}$
The bar chart above confirms the grid: the total 7 appears more often than any other total, which is exactly what the grid reveals when you count highlighted cells. Grids do not just answer one question — they let you read off probabilities for any total at a glance.
Grids work just as well when outcomes are not numbers. Suppose a bag contains a red (R), blue (B), and green (G) counter, and a spinner has sections labelled 1, 2, and 3. A table listing all counter-spinner combinations enumerates the full sample space of 3 × 3 = 9 outcomes without any guesswork.
| Spinner 1 | Spinner 2 | Spinner 3 | |
|---|---|---|---|
| Red (R) | R1 | R2 | R3 |
| Blue (B) | B1 | B2 | B3 |
| Green (G) | G1 | G2 | G3 |
From this table, questions like "what is the probability of picking a red counter and an odd number?" become a matter of counting: R1 and R3 are the favourable outcomes, giving $P = \frac{2}{9}$. The table has done the hard work of organisation; you supply the counting.
Tables and grids are ideal when two events produce independent outcomes. Venn diagrams take over when you need to show how groups of outcomes overlap — for example, which students study both French and Spanish, or which numbers are both even and multiples of 3.
A Venn diagram consists of a rectangle representing the universal set $\xi$ (everything under consideration) and circles inside it representing individual sets. Where two circles overlap, the overlapping region holds outcomes that belong to both sets simultaneously — the intersection $A \cap B$.
Four regions appear in any two-set Venn diagram: A only, B only, $A \cap B$ (both), and neither. Every outcome in the universal set must sit in exactly one of these four regions. If you place an outcome in two regions, you have made an error.
Worked example: Constructing a Venn Diagram from Given Information
In a class of 30 students, 18 study French (F), 14 study Spanish (S), and 7 study both. The rest study neither. Draw a Venn diagram and find $P(F \cup S)$.
Answer: $P(F \cup S) = \dfrac{5}{6}$. Venn diagram regions: F only = 11, both = 7, S only = 7, neither = 5.
Notice the key move in Step 2 and Step 3: always subtract the intersection before placing values in the individual circle regions. Placing the full totals (18 and 14) directly into the circles is the most common error on this type of question — it double-counts the 7 students who study both subjects.
Once a Venn diagram is correctly filled in, it becomes a probability machine. Every region holds a count, and every probability is that count divided by the total in $\xi$. The skill is translating the probability notation into the correct region or combination of regions.
The complement rule $P(A') = 1 - P(A)$ is particularly useful on exam papers because it turns a complicated region into a simple subtraction. If finding $P(A)$ is straightforward, finding $P(A')$ costs you almost no extra effort.
Both grids and Venn diagrams are tools for systematic enumeration — the same underlying idea expressed differently. A grid suits two-event experiments where you list every combination (rolling dice, spinning spinners, drawing from bags). A Venn diagram suits set-membership problems where outcomes can belong to overlapping categories.
On an Higher-tier paper, the question usually signals which tool to use. Phrases like "draw a sample space diagram" or "complete the table" point to a grid. Phrases like "complete the Venn diagram" or "the sets A and B are shown" point to Venn diagrams. If neither is specified, choose whichever makes counting easier for the specific problem.
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Move to the related lessons or back to the topic page when you need the wider sequence before exam-style practice.
A sample space diagram is a two-way grid listing every outcome of two combined events, with one event across the top and the other down the side. For two dice it has 6 × 6 = 36 cells, and 6 of them total 7, so P(total = 7) = 6/36 = 1/6.
The complement A′ is everything in the universal set ξ that is not in A, shown on a Venn diagram as everything outside circle A. Use P(A′) = 1 - P(A) to turn a complicated region into a simple subtraction.
The intersection A ∩ B is the overlapping region of the two circles, holding outcomes that belong to both sets. A two-set Venn diagram has four regions: A only, B only, A ∩ B and neither, and every outcome sits in exactly one of them.
Start with the intersection, then subtract it from each set total, then find the neither region. With 30 students, 18 studying French, 14 studying Spanish and 7 studying both, French only = 11, Spanish only = 7, both = 7 and neither = 5.
The union A ∪ B includes everything in A and everything in B, counting the intersection only once. With French only 11, both 7 and Spanish only 7 out of 30 students, P(F ∪ S) = 25/30 = 5/6.
The probability of a total of 7 with two fair dice is 1/6. A 6 by 6 grid gives 36 outcomes, and 6 of them total 7: (1,6), (2,5), (3,4), (4,3), (5,2) and (6,1), so P = 6/36 = 1/6.
The most likely total when rolling two dice is 7, which appears in 6 of the 36 cells of the sample space grid. The totals range from 2 (1 + 1) to 12 (6 + 6).
A table of 3 counters (red, blue, green) against a spinner showing 1, 2 and 3 gives 9 outcomes. Red with an odd number is R1 and R3, so the probability is 2/9.
For 30 students where 18 study French, 14 study Spanish and 7 study both, place 7 in the overlap first. Then French only is 18 - 7 = 11, Spanish only is 14 - 7 = 7, and neither is 30 - 25 = 5.
The universal set, ξ, is everything under consideration in a Venn diagram, shown as the rectangle. Every outcome must sit in exactly one region inside it, including the 'neither' region outside the circles.
A two-set Venn diagram has four regions: A only, B only, A ∩ B (both) and neither. Every outcome in the universal set sits in exactly one of these regions.
The most common error is placing the full totals straight into the circles instead of subtracting the intersection first. This double-counts the outcomes in both sets, such as the 7 students studying both subjects.
A sample space grid suits two-event experiments where every combination is listed, such as rolling dice. A Venn diagram suits set-membership problems where outcomes can belong to overlapping categories.
A ∪ B, the union, means everything in A, in B or in both, counting the intersection only once. For French and Spanish with regions 11, 7 and 7, the union is 25 out of 30, so P = 5/6.