GCSE Maths Practice: tree-diagrams

Independent Events in Probability

This question involves flipping a coin twice. Coin flips are one of the simplest examples of independent events in GCSE Maths. Independent events are events where the outcome of one event does not affect the outcome of the next event.

What Does “Independent” Mean?

Two events are independent if knowing the result of the first event gives you no information about the result of the second event. For a fair coin, every flip has the same probabilities, no matter what happened before. This means the probability of heads or tails stays the same on every flip.

The Multiplication Rule

When events are independent and happen in sequence, you can find the probability of both events happening by multiplying their probabilities:

P(A then B) = P(A) × P(B)

This rule works because the second probability does not change.

Using a Tree Diagram

A tree diagram can help visualise independent events. Each branch represents an outcome, and each branch has the same probability at every stage. For two coin flips, the probabilities on the second level of the tree are exactly the same as on the first level.

Worked Example 1 (Different Question)

A fair coin is flipped twice. Find the probability of getting two heads.

• P(first head) = 1/2
• P(second head) = 1/2
• Multiply: 1/2 × 1/2

Worked Example 2 (Another Independent Situation)

A spinner has four equal sections numbered 1, 2, 3 and 4. The spinner is spun twice. Find the probability of landing on an even number both times.

• P(even) = 2/4
• The probability stays the same on the second spin
• Multiply: 2/4 × 2/4

Independent vs Dependent Events

It is important not to confuse independent events with dependent events. Drawing items from a bag without replacement is dependent, because the probabilities change. Coin flips, dice rolls, and spinners are independent because each outcome resets.

Common Mistakes

• Adding probabilities instead of multiplying
• Assuming probabilities change when they do not
• Confusing independent events with “without replacement” problems

Real-Life Examples

Independent probability appears in many real-life situations, such as rolling dice in board games, spinning wheels in gameshows, or generating random numbers in computer programs.

Mini FAQ

• Does a coin remember previous flips? No. Each flip is independent.
• Do probabilities ever change for coins? Only if the coin is biased.
• Should I always draw a tree diagram? It helps, but for simple independent events it is not always required.

Study tip: If the probabilities stay the same at every stage, the events are independent.