GCSE Maths Practice: tree-diagrams

Two-Step Probability Without Replacement (GCSE Foundation)

Many GCSE probability questions involve making two selections one after another. The key skill is deciding whether the second probability stays the same or changes. If the question says without replacement, the second probability changes because one item has been removed from the set.

Dependent Events: What It Means

When the outcome of the first selection affects the second selection, the events are called dependent events. This is common when you take something from a group (like cards, sweets, stickers, or tokens) and do not put it back before choosing again. The total number of items decreases, and sometimes the number of favourable outcomes changes too.

Method: “First, Then, Multiply”

For a sequence such as “Green then Blue”, you can use this rule:

P(Green then Blue) = P(Green) × P(Blue given Green)

The phrase “given Green” means you adjust the second probability based on what happened in the first step.

How Tree Diagrams Help

A tree diagram is a neat way to organise the probabilities. The first set of branches shows the possible outcomes of the first selection. From each branch, you can show the possible outcomes of the second selection. To find the probability of a particular sequence, you multiply the probabilities along that path.

Worked Example 1 (Different Context)

A box contains 5 football cards and 3 basketball cards. Two cards are picked without replacement. Find the probability of picking a football card then a basketball card.

• P(football first) = 5/8
• After removing a football card, total left = 7 and basketball cards still = 3
• P(basketball second | football first) = 3/7
• Multiply: 5/8 × 3/7

Worked Example 2 (Spot the Common Mistake)

Suppose a jar contains 4 red counters and 6 yellow counters. Some students write P(red then red) as (4/10) × (4/10). This is incorrect without replacement because after taking a red counter there are only 3 reds left out of 9 total. The second fraction must change when the item is not returned.

Common Mistakes to Avoid

• Not changing the denominator: the total becomes one less on the second selection.
• Confusing “then” with “or”: “then” usually means multiply, while “or” often means add (with care).
• Using equivalent answers twice: two fractions like 2/18 and 8/72 are the same value, so only one should appear as an option.
• Expanding unnecessary branches: for some GCSE Foundation questions, you only need to expand the branch that leads to the required path.

Real-Life Link

This type of probability is used in real life when sampling items from a batch (quality control), picking random winners from a set of entries, or drawing items from a collection without returning them. The big idea is always the same: once something is removed, the situation changes.

Mini FAQ

• When do probabilities stay the same? When there is replacement, or the events are independent.
• Do I have to use a tree diagram? Not always, but it is one of the clearest methods for two-step GCSE questions.
• Should I simplify my final fraction? Often yes, but many mark schemes accept an equivalent unsimplified fraction.

Study tip: If you see “without replacement”, immediately think: “the second fraction will use a denominator that is one smaller.”