Key Stage 4 maths · Probability

GCSE probability: tree diagrams and combined events explained

GCSE probability goes beyond one-step questions about dice and coins. Children use the fact that the probabilities of all possible, mutually exclusive outcomes add up to 1, use a probability model to predict what will happen in future experiments, and calculate the probability of two or more events happening together using tree diagrams. They also learn to state the assumptions a model depends on, such as whether items are replaced.

Tree diagrams and combined events, explained

Two events are mutually exclusive if they can’t happen at the same time, like rolling a 2 and rolling a 5 on one throw. A set of events is exhaustive if it covers every possible outcome. When a set of events is both, their probabilities add up to exactly 1. This is why P(not A) = 1 − P(A), and why a table of probabilities for a biased spinner must total 1, with a missing value found by subtraction. Probabilities are written as fractions, decimals or percentages, and always lie between 0 and 1 (or 0% and 100%).

A probability model is a set of probabilities used to predict outcomes. Some models are theoretical, such as 1/6 for each face of a fair dice. Others come from data, where the relative frequency (the number of times something happened divided by the number of trials) is used as an estimate. Children learn that a model can be used to predict the expected number of times an outcome occurs: if the probability of red on a spinner is 0.3, then in 200 spins we expect about 0.3 × 200 = 60 reds. The real result is rarely exactly 60. They also learn that relative frequency from an unbiased sample tends to move closer to the theoretical probability as the number of trials increases, but this does not mean that results “even out” in the short term.

To find the probability of two or more events together, children first decide whether the events are independent, meaning one does not affect the other, or dependent, meaning the first changes the probabilities for the second. For independent events, P(A and B) = P(A) × P(B). For dependent events, the second probability is changed by what happened first, as when counters are taken from a bag and not put back. A tree diagram shows every route: probabilities are written on the branches, multiplied along each route for “and”, and added across different routes for “or”. It is often quicker to find “at least one” by working out the probability of none and subtracting from 1.

Every answer rests on assumptions, and GCSE questions often ask pupils to state them. The coin is fair, the choices are random, the events are independent, and the item is or isn’t replaced. A good answer notices when real life breaks them, for example that wet days tend to follow wet days, so “independent daily weather” is only a rough model.

Words your child will hear

Mutually exclusive: Events that can’t happen at the same time, such as rolling an even number and rolling a 5.
Exhaustive: A set of events that together cover every possible outcome.
Independent events: Events where the result of one does not change the probability of the other.
Dependent events: Events where the first result changes the probability of the second, as with no replacement.
Relative frequency: How often something happened divided by the number of trials, used as an estimate of probability.
Tree diagram: A diagram of branches showing every possible route through two or more events, with probabilities on each branch.
Expected frequency: The number of times an outcome is predicted: the probability multiplied by the number of trials.

What the National Curriculum says

In Key Stage 4, pupils should be taught to:

Apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to 1
Use a probability model to predict the outcomes of future experiments; understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size
Calculate the probability of independent and dependent combined events, including using tree diagrams and other representations, and know the underlying assumptions

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

A bag holds 5 red counters and 3 blue counters. Two counters are taken out at random, one after the other, without replacing the first. Find the probability that both are the same colour.

There are 8 counters, so the first pick has probabilities 5/8 for red and 3/8 for blue. The counter isn’t replaced, so the second pick is from 7 counters. This is a dependent event.
Draw the tree. If the first is red, 4 red and 3 blue remain out of 7. If the first is blue, 5 red and 2 blue remain out of 7. Multiply along each branch.
```
1st         2nd         Route
R (5/8) --> R (4/7)   20/56
        --> B (3/7)   15/56
B (3/8) --> R (5/7)   15/56
        --> B (2/7)    6/56
```
Both the same colour means red then red, or blue then blue. These are different routes, so add them: 20/56 + 6/56 = 26/56.
Simplify: 26/56 = 13/28, which is about 0.46.
Check: the other routes (different colours) are 15/56 + 15/56 = 30/56. Since 26/56 + 30/56 = 56/56 = 1, the probabilities of the exhaustive set of outcomes sum to 1.

The probability that both counters are the same colour is 13/28 (about 0.46).

Common mistakes and how to help

Adding along the branches and multiplying between routes

“And” and “or” are easy to mix up when there are many numbers on a diagram. A route needs the first event and the second, so those are multiplied. Different routes are alternatives, so they are added.

What helps: Say each route in words: “red and then red”. A route is multiplied. When the question allows more than one route, say “or” and add. A quick check is that probabilities of all routes should total 1.

Keeping the same probabilities on the second branches when nothing is replaced

In independent problems (coins, spinners, dice) the second set of branches is the same as the first, so the habit of copying them is natural.

What helps: Ask what is in the bag now. After a red is removed, there are fewer counters, and fewer reds. The denominator drops by one and the numerator drops if the colour matches. Ask “with or without replacement?” of every question.

Believing a run of results makes the opposite more likely

After five heads, it feels as if tails is “due”. The coin has no memory: each toss is independent, so the next toss is still 1/2. Large samples settle towards the theoretical probability because the early run is swamped by many more trials, not because the coin corrects itself.

What helps: Do the experiment. Flip a coin 20 times and 200 times, record the running relative frequency, and see how the longer run settles near 1/2 while the first few flips are wild.

Assuming outcomes are equally likely without checking

“It will either rain or not, so 50%.” Two outcomes aren’t necessarily equally likely. Equal chances only come from a fair, symmetrical situation or from data.

What helps: Ask whether there is a reason for outcomes to be equally likely, or whether the probabilities came from data. If probabilities are given, check they add to 1 and calculate the missing one by subtraction.

How to help at home

Run the long experiment. Flip a coin or roll a dice 100 times between you, tallying in groups of ten. Work out the relative frequency after 10, 50 and 100 trials and compare it with the theoretical probability. It shows why larger samples are more reliable, and why short runs mislead.
Socks in a drawer. Put 4 black and 2 white socks in a bag. Ask for the probability that two taken out are both black: 4/6 × 3/5 = 2/5. Then draw two socks 30 times, putting both back each time, and see how close your tally gets.
Draw a tree for a family decision. Pick two simple events with known chances, such as a coin flip deciding the film and a dice roll deciding the snack. Draw all the branches together, multiply, and check that all the routes add up to 1.
Hunt for assumptions. When a forecast or a news story quotes a chance of something, ask what has to be true for that number to be right. Are the events independent? Is the sample fair? Naming assumptions is a GCSE skill and a useful everyday one.

Builds on

Key Stage 3: Probability and Venn diagrams