GCSE Maths Practice: probability-scale

Understanding Independent Probability with Dice Rolls

This question involves calculating the probability of two specific outcomes occurring one after the other when rolling a fair six-sided die twice. Each roll is an example of an independent event, meaning the result of the first roll does not influence the outcome of the second. Independence is a central idea in probability and is vital in GCSE Maths for analysing multi-step scenarios, interpreting tree diagrams, and modelling real-world situations involving repeated random processes.

A standard fair die has six faces labelled 1 to 6. Each face has an equal chance of landing face-up, so the probability of rolling any particular number is \(1/6\). When rolling the die twice, we consider each roll separately. Because neither roll depends on the other, the probability of both events happening is calculated by multiplying their individual probabilities. This multiplication rule is essential for all independent sequential events.

Interpreting the Problem

The question asks for the probability of rolling a 6 first and then a 4 on the second roll. Since order matters, the outcome (6,4) is different from outcomes like (4,6) or (6,1). When order is specified, each pair of results forms a distinct combined outcome. With two rolls, there are 36 possible combined outcomes (6 possibilities for the first roll and 6 for the second), all equally likely. This makes it easy to visualise why each specific pair of outcomes has a probability of \(1/36\).

Step-by-Step Method

1. Calculate the probability of the first event: rolling a 6 has probability \(1/6\).
2. Calculate the probability of the second event: rolling a 4 also has probability \(1/6\).
3. Multiply the probabilities: because the events are independent, the combined probability is \((1/6)(1/6) = 1/36\).

This method works for any pair of independent events involving dice, coins, spinners, or any random processes that do not influence each other.

Worked Example 1

What is the probability of rolling two even numbers in a row? Even numbers on a standard die are 2, 4, and 6—three favourable outcomes. So the probability of rolling an even number is \(3/6 = 1/2\). For two rolls: \((1/2)(1/2) = 1/4\).

Worked Example 2

What is the probability of rolling a 1 followed by any number greater than 4? The first probability is \(1/6\). Numbers greater than 4 are 5 and 6, so the second probability is \(2/6 = 1/3\). Combined probability = \((1/6)(1/3) = 1/18\).

Common Mistakes

• Adding probabilities instead of multiplying them.
• Thinking the second roll becomes more or less likely after the first roll—this is incorrect for independent events.
• Confusing order-specific and non-order-specific questions.
• Miscounting total possible outcomes when visualising the 6 × 6 grid of results.

Real-Life Applications

These ideas extend far beyond dice. Independent probability is used in genetics (predicting independent allele combinations), computing (random number generation), manufacturing (defect testing), and financial modelling. Any time multiple unrelated random events occur in sequence, independence plays a key role in determining likely outcomes.

FAQ

Q: Why do we multiply, not add?
A: Multiplication is used for 'and' events when both must happen in sequence.

Q: Does the order of events matter?
A: Yes. Rolling (6,4) is different from (4,6) unless the question says order does not matter.

Q: Are dice always independent?
A: Yes—each roll is unaffected by previous rolls unless the problem states otherwise.

Study Tip

Whenever dealing with repeated independent events, build the habit of multiplying probabilities. A quick sketch of the 6 × 6 grid (sample space) can also help you confirm your answer visually.