Sample Space Diagrams

A sample space diagram shows every possible outcome when two things happen together, such as throwing a coin and spinning a spinner. Some exam papers call it a possibility space diagram.

In the exam you are often asked to list all the possible combinations, for 2 marks. You may also be asked to complete a sample space diagram, for 1 mark, and then use it to work out a probability, for 1 or 2 marks.

Harder questions ask which of two outcomes is more likely. Then you must work out both probabilities and compare them.

To list every combination, keep the first choice fixed and go through every second choice in order. Then move on to the next first choice and do the same again. This stops you missing or repeating a combination.

A grid does this for you. Put one event down the side and the other across the top. Each cell inside the grid is one outcome: one result from the side together with one result from the top.

Every row has the same number of cells, so you can check your count:

\text{number of outcomes} = \text{first choices} \times \text{second choices}

Try it: the slider labelled First-stage choices sets how many numbers the first spinner has (the rows). The slider labelled Second-stage choices sets the second spinner (the columns). In each cell, the first number is the first spinner and the second number is the second spinner. Count the cells each time: it is always rows times columns. Set both sliders to 6 to see the 36 outcomes for two ordinary dice.

When every cell is equally likely, count the cells that fit and divide by the total number of cells:

\text{probability} = \frac{\text{number of cells that fit}}{\text{total number of cells}}

The cells are equally likely when the coins, dice and spinners are fair and every sector of a spinner is the same size. If a coin is biased, or a spinner has one big sector, the cells are not equally likely and this method does not work.

If a fair spinner has four equal sectors labelled 1, 2, 2 and 3, give each sector its own row. The number 2 then appears in two rows, because it is twice as likely as 1 or 3.

Often the two numbers are added or multiplied to make a score. Each cell then shows the side number plus the top number, or the side number times the top number. A + or \times in the corner of the grid reminds you which.

Different cells can show the same score. Spinner 1 with dice 3, and spinner 3 with dice 1, both give a total of 4, but they are different outcomes. Count every cell, not just the different scores.

• Repeating or missing combinations. Keep one choice fixed, go through every other choice in order, and check the count is first choices times second choices.
• Writing combinations that run together. Write each outcome as its own pair, such as BW or (2, 5), with a clear gap between pairs.
• Dividing by the number of different totals. Two dice give 36 equally likely cells, so divide by 36, not by the 11 different totals.
• Assuming every outcome is equally likely. Count cells only when the coins, dice or spinners are fair and the sectors are equal.
• Comparing counts instead of probabilities. Write each probability as a fraction, with its own denominator, before you compare.
• Misreading “more than” and “or more”. “More than 6” leaves out 6, but “6 or more” includes it.