Construct theoretical possibility spaces for single and combined…

The idea

A possibility space lists every equally likely outcome. For a single experiment, such as rolling one fair six-sided die, the space is simply 1, 2, 3, 4, 5, 6, so the probability of an even number is 3/6 = 1/2. For two combined experiments, draw a grid: put one die down the side and the other along the top, then write the total in each of the 36 cells. Every cell is equally likely, so a probability is the number of cells you want divided by 36. Four cells hold a total of nine, so P(total 9) = 4/36 = 1/9. This method depends on the cells being equally likely. The totals themselves are not equally likely, which is exactly why you count cells rather than count the different totals available.

What earns the marks

• A correctly built possibility space for a single experiment, listing every equally likely outcome once.
• A correctly built grid for a combined experiment, with every combination appearing once.
• Identifying the outcomes that satisfy the condition, shown by ringing or listing them.
• A probability whose denominator is the total number of equally likely outcomes.
• The answer in its simplest form when the question asks for it.

Common mistakes and how to put them right

• Using the number of different totals as the denominator, for example 11 rather than 36 for two dice.

• Counting a pair such as one and three only once when the two dice are distinguishable, so the count comes out short.

• Adding two probabilities when the question needs both events to happen together.

• Writing the answer as a ratio of wanted cells to unwanted cells instead of as a probability.

Exam tips

• Draw the grid even when the question does not ask for one. The probability marks come faster off a grid than off a written list.
• Ring the cells you are counting rather than counting them in your head, so you can recount them if the number looks wrong.