GCSE Maths Practice: tree-diagrams

Independent Events in Probability

This question involves two different actions: flipping a coin and rolling a dice. These are examples of independent events. Independent events are events where the outcome of one event does not affect the outcome of the other.

Why These Events Are Independent

When you flip a fair coin, the result is either heads or tails, each with the same probability. When you roll a fair six-sided dice, each number from 1 to 6 has an equal chance of appearing. The coin does not influence the dice, and the dice does not influence the coin. This is why the events are independent.

The Multiplication Rule

For independent events that happen together, the probability is found using:

P(A and B) = P(A) × P(B)

This rule applies whenever two events are independent and both must happen.

Using a Tree Diagram

A tree diagram is a useful way to organise independent events. Each stage shows the possible outcomes and their probabilities. For independent events, the probabilities on the second stage stay the same regardless of the outcome of the first stage.

Worked Example 1 (Different Question)

A fair coin is flipped and a spinner with four equal sections is spun. Find the probability of getting tails and landing on an even number.

• P(tails) = 1/2
• P(even) = 2/4
• Multiply: 1/2 × 2/4

Worked Example 2 (Another Independent Situation)

A dice is rolled and then a card is chosen from a standard deck. Find the probability of rolling a 6 and then choosing a heart.

• P(6) = 1/6
• P(heart) = 13/52
• Multiply: 1/6 × 13/52

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 event resets.

Common Mistakes

• Adding probabilities instead of multiplying
• Assuming probabilities change when they do not
• Using the wrong probability for the dice or coin

Real-Life Examples

Independent probability appears in games, lotteries, simulations, and computer programming where random number generators are used. Each event is separate and does not depend on previous outcomes.

Mini FAQ

• Does the dice remember previous rolls? No. Each roll is independent.
• Does a coin affect a dice roll? No. They are completely separate events.
• Do I always need a tree diagram? Not always, but it helps visualise multi-step problems.

Study tip: If two actions are unrelated and random, they are usually independent and their probabilities should be multiplied.