Formal Probability Rules
Formal probability rules provide systematic methods for calculating the likelihood of combined events. The addition rule determines probabilities for 'A or B' scenarios, whilst the multiplication rule handles 'A and B' situations. These rules form the foundation for analysing complex probability problems in Year 9 and GCSE mathematics.
Why it matters
Formal probability rules underpin risk assessment in insurance, where actuaries calculate premiums by combining multiple risk factors. Weather forecasters use these principles to determine the probability of rain occurring on consecutive days, applying independence assumptions. In medicine, diagnostic tests combine probabilities from multiple symptoms to assess disease likelihood. Quality control in manufacturing relies on these rules to calculate failure rates when multiple components must function together. Sports betting odds reflect sophisticated probability calculations using addition and multiplication rules. The complement rule appears in reliability engineering, where P(system failure) = 1 - P(system works). These concepts prepare students for A-level statistics and university-level probability theory, forming essential groundwork for careers in data science, finance, and research.
How to solve formal probability rules
Probability — Addition & Multiplication Rules
- Addition rule (OR): P(A or B) = P(A) + P(B) − P(A and B).
- If mutually exclusive: P(A or B) = P(A) + P(B).
- Multiplication rule (AND, independent): P(A and B) = P(A) × P(B).
- Use tree diagrams to organise compound events.
Example: Two coins: P(HH) = 12 × 12 = 14.
Worked examples
P(A) = 0.75. Find P(not A).
Answer: 0.25
- Apply complement rule → P(not A) = 1 - P(A) = 1 - 0.75 = 0.25 — The complement rule: P(not A) = 1 - P(A).
P(A) = 16, P(B) = 14, A and B are mutually exclusive. P(A or B)?
Answer: 512
- Apply addition rule for mutually exclusive events → P(A or B) = P(A) + P(B) = 16 + 14 — When events cannot happen together, add their probabilities.
- Calculate → 16 + 14 = 512 — Find a common denominator and add.
P(rain) = 0.5 each day. P(no rain both days) if independent?
Answer: 0.25
- Find P(no rain) for one day → P(no rain) = 1 - 0.5 = 0.5 — Use the complement rule.
- Multiply for independent events → P(no rain both) = 0.5 x 0.5 = 0.25 — For independent events, multiply the individual probabilities.
Common mistakes
- Adding probabilities incorrectly when events overlap, such as calculating P(A or B) = 0.4 + 0.3 = 0.7 instead of applying the general addition rule P(A or B) = 0.4 + 0.3 - 0.1 = 0.6 when P(A and B) = 0.1.
- Multiplying dependent events as if independent, computing P(second card is heart | first was heart) × P(first is heart) = 12/51 × 13/52 instead of recognising the conditional nature of card drawing without replacement.
- Forgetting the complement relationship, writing P(not A) = 0.3 when P(A) = 0.6 instead of correctly calculating P(not A) = 1 - 0.6 = 0.4 using the complement rule.