IGCSE Maths / Probability

Theoretical probability and sample space revision notes

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

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A sample space grid shows every possible outcome when two events happen together. You can use it to find probability by counting equally likely outcomes, then simplifying the fraction.

How a sample space grid works

A sample space is a list or diagram of every possible outcome. For two dice, put the first die's scores down the side and the second die's scores across the top. Each box pairs one result from each die. The number of boxes is the number of rows multiplied by the number of columns. With two dice, there are 6 × 6 = 36 boxes. Each box is equally likely because both dice are fair.

How to find a probability from the grid

Count the boxes that match the outcome you want, then compare that count with the total number of boxes. The boxes are the equally likely outcomes, so they belong underneath the fraction. For a total of 7, six boxes match, so the probability is 6/36. For an even total, count the boxes containing 2, 4, 6, 8, 10 or 12. There are 18, so the probability is 18/36, which is half.

How to simplify the probability

A probability fraction can be simplified without changing its value. Divide the numerator and denominator by the same number.

Why you count boxes, not totals

The possible totals from two dice are 2 to 12, but those totals are not equally likely. A total of 2 can happen in only one way, while a total of 7 can happen in six ways. Therefore, the 11 totals cannot be used as the equally likely outcomes. For a coin and a die, the grid has 2 rows and 6 columns, giving 2 × 6 = 12 outcomes. For a total of 12 with two dice, only the outcome 6 and 6 works, so the probability is 1/36.

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