Listing outcomes systematically revision notes

Edexcel International GCSE Maths (4MA1), chapter 20: Probability, specification point 6.3c-f. Free lesson.

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A systematic list records every possible outcome of a combined event without missing or repeating any. Count the outcomes first, then use an ordered list or grid to find probabilities reliably.

How a grid lists every outcome

For a combined event, each outcome pairs one result from the first event with one result from the second. Put one set of results down the side and the other across the top. Each box is one outcome, so the grid makes the full list clear. A coin has 2 results, H and T. A four-sided spinner has 4 results, 1 to 4. The grid has 2 rows and 4 columns, so there are 2 × 4 = 8 outcomes. A head appears in all 4 boxes in the top row. Therefore, P(head) = 4/8 = 1/2.

How to check the number of outcomes

Work out how many outcomes there should be before you start listing. For each choice of the first thing, there are all the choices of the second thing. This gives the total by multiplication, and it tells you when your list is complete. For 3 starters and 4 mains, there are 4 mains for each starter. So the number of two-course meals is 3 × 4 = 12. If a list has only 11 meals, you know one is missing.

How to list outcomes in a fixed order

A list written in whatever order results come to mind is hard to check. You may stop when you run out of ideas, even though an outcome is missing. Instead, hold one result fixed and run through every possibility for the other result. Then change the fixed result and repeat. For the coin and spinner, keep H fixed and list 1, 2, 3, 4. Then keep T fixed and list 1, 2, 3, 4. This method gives every pairing once, and the total can be checked against the number you worked out first. A grid is a clear way to make this systematic list.

Why order matters for different dice

When two dice can be told apart, their results are ordered pairs. The result on the first die is not interchangeable with the result on the second die, so switching the numbers gives a different outcome. A 3 on the first die and a 5 on the second is different from a 5 on the first and a 3 on the second. There are 6 choices for each die, giving 6 × 6 = 36 outcomes, not 21. Use the correct total number of outcomes when finding the denominator of a probability.

How to count combinations of choices

The same multiplication method works when choices are grouped in different ways. Count the choices for each feature, then multiply to find the number of possible combinations. With 5 shirt colours and 3 sizes, there are 5 × 3 = 15 different shirts. Remember to count outcomes first, multiply the choices, and use a fixed order if you write them all out.