Probability for GCSE – From Basics to Advanced

Introduction

Probability measures the likelihood of an event occurring, and it’s a key topic in GCSE Maths. From basic concepts to advanced techniques, understanding probability is crucial for success.

This article will cover:

1. Basic probability rules.
2. Tree diagrams and independent events.
3. Real-life applications and exam tips.

1. Basic Probability Rules

Key Formula

Probability=Number of Favourable OutcomesTotal Number of Outcomes\text{Probability} = \frac{\text{Number of Favourable Outcomes}}{\text{Total Number of Outcomes}}Probability=Total Number of OutcomesNumber of Favourable Outcomes​

Example: What is the probability of rolling a 3 on a fair six-sided die?

• Favourable outcomes: 1 (rolling a 3).
• Total outcomes: 6.
• Probability = 16.\frac{1}{6}.61​.

Complementary Events

The probability of an event not happening is:

P(Not A)=1−P(A).P(\text{Not A}) = 1 – P(\text{A}).P(Not A)=1−P(A).

Example: If P(A)=0.7P(\text{A}) = 0.7P(A)=0.7, then P(Not A)=0.3.P(\text{Not A}) = 0.3.P(Not A)=0.3.

2. Tree Diagrams

Tree diagrams help visualise probabilities for sequential events.

Example: A bag contains 3 red and 2 blue balls. If two balls are drawn (without replacement), what’s the probability of drawing one red and one blue?

1. Draw branches for each draw.
2. Calculate probabilities:
   • First red, then blue: 35×24=620.\frac{3}{5} \times \frac{2}{4} = \frac{6}{20}.53​×42​=206​.
   • First blue, then red: 25×34=620.\frac{2}{5} \times \frac{3}{4} = \frac{6}{20}.52​×43​=206​.
3. Total probability: 620+620=1220=0.6.\frac{6}{20} + \frac{6}{20} = \frac{12}{20} = 0.6.206​+206​=2012​=0.6.

3. Real-Life Applications

• Weather Forecasting: Predicting rain or sunshine.
• Games of Chance: Analysing odds in games like dice or cards.
• Risk Analysis: Understanding probabilities in business or insurance.

Practice Question

Question: A coin is flipped twice. What is the probability of getting at least one head?
Solution:

1. Complementary event: Both tails = 12×12=14.\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}.21​×21​=41​.
2. P(At least one head)=1−14=34.P(\text{At least one head}) = 1 – \frac{1}{4} = \frac{3}{4}.P(At least one head)=1−41​=43​.

Conclusion

Probability connects mathematics with real-life decision-making. Regular practice with tree diagrams and complex events will help you solve probability problems confidently in GCSE Maths.