Sample Spaces

A sample space is the list of all possible outcomes of an experiment. For a single die it is just 1, 2, 3, 4, 5, 6. But often two things happen at once — two dice, or a coin and a die — and we need every combination.

A two-way sample space diagram (a grid) lists them all: one event runs down the rows, the other across the columns, and each cell is one combined outcome. Two dice fill a 6 \times 6 grid, so there are

6 \times 6 = 36 \text{ outcomes.}

To work out a probability when two events happen together:

list every possible outcome — don't miss any;
a grid (two-way diagram) handles two events neatly, one per axis;
the total number of outcomes is (\text{ways for the first}) \times (\text{ways for the second});
then, exactly as before, P(\text{event}) = \frac{\text{favourable outcomes}}{\text{total outcomes}}.

The grid for two dice

Die A runs down the rows, die B across the columns. Step through to fill in the sums, then to spot every way of making a total of 7.