Probability of combined events

Combined events

A combined event involves two or more things happening: two dice rolled together, a coin thrown and a spinner spun, two counters taken from a bag one after the other, or the weather on two different days.

There are three tools for working out these probabilities, and part of the skill is choosing the right one:

• a sample space diagram, when two experiments each have a small number of equally likely outcomes;
• a tree diagram, when the outcomes are not equally likely or there are several stages;
• a Venn diagram, when people or items are sorted into two overlapping groups.

On the Core papers combined events are always with replacement, or independent: what happens first does not change the probabilities for what happens next. Extended questions may also be without replacement.

Sample space diagrams

A sample space diagram lists every possible outcome of two experiments in a grid: the outcomes of one experiment along the top, the other down the side, and each cell shows one combination.

If each experiment is fair, every cell is equally likely. So

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

Two fair dice give $6\times 6=36$ cells. A fair coin and a fair dice give $2\times 6=12$. The cells can show a total, a difference, a product or just the pair of results, such as (H, 4).

Why not just list the totals? With two dice the totals 2 to 12 are not equally likely. A total of 2 happens only one way, (1, 1), but a total of 7 happens six ways. Counting cells takes this into account; counting totals does not. Note too that (2, 5) and (5, 2) are different cells: the first dice shows 2 in one and 5 in the other.

Independent events: multiply for "and"

Two events are independent when one happening does not change the probability of the other. Two dice, a coin and a spinner, and a counter that is put back before the next is taken are all independent.

For independent events,

$$\text{probability of}A\text{and}B=\text{probability of}A\times \text{probability of}B$$

Why multiply? Think of a large number of trials. A six comes up on the first dice in $\frac{1}{6}$ of them. Of those trials, the second dice shows a six in $\frac{1}{6}$. So both are sixes in $\frac{1}{6}$ of $\frac{1}{6}$ of the trials: $\frac{1}{6}\times \frac{1}{6}=\frac{1}{36}$, exactly what the sample space shows (one cell of 36).

For outcomes that cannot happen together, such as "red then blue" and "blue then red", the probability of one or the other is found by adding. So: multiply along a path for "and"; add different paths for "or".

Tree diagrams

A tree diagram shows each stage as a set of branches.

• The outcomes are written at the end of each branch.
• The probabilities are written beside each branch.
• The branches leaving any one point cover every possibility, so their probabilities add up to 1.

To use it:

1. Find each path that gives the event you want.
2. Multiply the probabilities along each path.
3. Add the results for the different paths.

When the first item is replaced, the second set of branches is the same as the first. A useful check: the probabilities at the ends of all the paths add up to 1.

"At least one" and the shortcut through 1

"At least one" means one or more. With two or three stages there are many paths that have at least one success, but only one path has none. It is quicker and safer to use

$$\text{probability of at least one}=1-\text{probability of none}$$

A very common wrong method is to add the single probabilities. For two days each with probability 0.3 of rain, $0.3+0.3=0.6$ is wrong, because it counts the case of rain on both days twice. With four such days, adding would give 1.2, which is impossible.

Venn diagrams for combined events

A Venn diagram with two sets shows two events at once. Each region can hold a number of people or a probability.

• "$A$ and $B$" is the overlap.
• "$A$ or $B$" means in $A$, in $B$, or in both: everything inside either circle.
• "$A$ but not $B$" is the part of $A$ outside the overlap.
• "Neither" is the region outside both circles.

All four regions together make the whole: they add to the total number, or to 1 for probabilities. That is how a missing value is found. On the Core papers Venn diagrams have two sets.