Two-stage experiments: the outcome table

A two-stage experiment is an experiment done in two stages, for example:

• two dice are thrown;
• a coin is tossed, then a die is thrown;
• a spinner is spun twice.

An outcome of such an experiment is a pair: the outcome of the first stage and the outcome of the second. It is written in brackets: $\displaystyle (2,5)$ is 2 in the first stage and 5 in the second.

• The order matters: $\displaystyle (2,5)$ and $\displaystyle (5,2)$ are two different outcomes.
• When two dice are thrown together, one of them is chosen to be “the first” – a blue die and a red die, say.

The outcome table shows all the outcomes of a two-stage experiment, a cell for each. Here a blue die and a red die are thrown:

• Each row is an outcome of the blue die, each column an outcome of the red die.
• Each cell is one outcome of the experiment: row 2, column 5 holds $\displaystyle (2,5)$.
• 6 rows and 6 columns: $\displaystyle {6\cdot6}={36}$ cells, so 36 outcomes.

When each stage’s outcomes are equally likely, all the cells of the table are equally likely too. So an event’s probability is found as in a one-stage experiment – by counting cells:

• $\displaystyle k$ – the number of cells in the event $\displaystyle A$;
• $\displaystyle n$ – the number of all the cells.

The example below finds the probability that the two dice add up to 8. Each cell shows the sum, and the event’s cells are marked:

• The event has 5 cells: $\displaystyle (2,6)$, $\displaystyle (3,5)$, $\displaystyle (4,4)$, $\displaystyle (5,3)$, $\displaystyle (6,2)$.
• The probability that the sum is 8: $\displaystyle \frac{5}{36}$.

The sum 7 has the most cells – 6 – so it is the likeliest sum. Its cells are marked:

• The possible sums are 2 to 12.
• The cells of each sum lie on one diagonal.
• The sum 2 has a single cell, $\displaystyle (1,1)$, and so does 12.
• The probability of a sum of 7: $\displaystyle {\frac{6}{36}}={\frac{1}{6}}$.

The example below finds the probability of heads and an even number, when a coin is tossed and a die thrown. The coin has 2 outcomes and the die 6, so the table has $\displaystyle {2\cdot6}={12}$ cells:

• The event has 3 cells: heads and 2, heads and 4, heads and 6.
• The probability: $\displaystyle {\frac{3}{12}}={\frac{1}{4}}$.

A blue die and a red die are thrown.
In each part, mark the cells of the event in the table, and find its probability.

Both dice show the same number.

At least one of the dice shows 6.

The blue die shows more than the red die.

A blue die and a red die are thrown.
Write the product of the two numbers in each cell.
Mark the cells of the event “the product is 12”, and find its probability.

A spinner has three equal parts, numbered 1, 2, 3.
The arrow is spun twice.
Write the sum of the two outcomes in each cell.
Mark the cells of the event “the sum is 5”, and find its probability.
Which sum is the likeliest?

A coin is tossed and a die is thrown.
Mark the cells of the event “the coin shows heads, or the die shows 6”, and find its probability.