GCSE Maths Practice: tree-diagrams

Sequential Probability Without Replacement

This GCSE Foundation question involves selecting two items one after another from the same group, where the first item is not replaced before the second is chosen. When this happens, the probabilities in the second step depend on what happened in the first step. These are known as dependent events.

Understanding the Key Phrase: “Without Replacement”

The phrase “without replacement” tells you that once an item is chosen, it is removed from the group. This changes the total number of items available and sometimes also changes how many favourable outcomes remain. As a result, the second probability must be calculated using updated numbers.

Method: First, Then, Multiply

For events written as “A then B”, the probability is found using:

P(A then B) = P(A) × P(B given A)

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

Worked Example 1 (Different Numbers)

A pencil case contains 4 blue crayons and 6 yellow crayons. Two crayons are taken one after another without replacement. Find the probability that the first crayon is blue and the second is yellow.

• P(blue first) = 4/10
• After a blue is taken, total left = 9 and yellow crayons still = 6
• P(yellow second | blue first) = 6/9
• Multiply: 4/10 × 6/9

Worked Example 2 (Spot the Common Error)

A common mistake is forgetting to change the denominator on the second probability. For example, if a jar has 3 red sweets and 7 green sweets, some students incorrectly write P(red then green) as (3/10) × (7/10). This is wrong without replacement because the total becomes 9 after the first sweet is taken.

Tree Diagrams and Why They Help

A tree diagram shows each stage of the probability clearly. Each branch represents a possible outcome, and the probability is written along the branch. To find the probability of a particular sequence, you multiply the probabilities along that path.

Common Mistakes to Avoid

• Using the original total for the second draw
• Adding probabilities instead of multiplying them
• Expanding unnecessary branches for Foundation questions
• Arithmetic errors when simplifying fractions

Real-Life Applications

Sequential probability is used when selecting items for quality checks, choosing winners in competitions, or drawing cards or tickets. The key idea is always the same: once something is removed, the situation changes.

Mini FAQ

• Do I need to draw the full tree? Not always. At Foundation level, it is usually enough to expand only the branch you need.
• Should I simplify my final answer? Yes, unless the question specifically says otherwise.
• When do probabilities stay the same? Only when items are replaced or events are independent.

Study tip: Whenever you see “then” and “without replacement”, think: “multiply, and reduce the total for the second step.”