Probabilities of an exhaustive set sum to 1
OCR J560 Foundation routinely sets "the probabilities of getting red, blue, green are…; find the probability of yellow." It's a 1–2 mark question that rewards a single subtraction.
The fundamental rule
For any random experiment, the set of all possible outcomes is called the sample space. The probabilities of all outcomes in the sample space sum to 1.
Σ P(outcome) = 1.
This is also called the complement rule when applied to one event:
P(event) + P(NOT event) = 1.
So:
P(NOT A) = 1 − PA.
When to use
OCR question patterns:
- "The spinner shows 1, 2, 3 or 4. P(1)=0.2, P(2)=0.3, P(3)=0.1. Find P(4)." → 1 − 0.2 − 0.3 − 0.1 = 0.4.
- "P(rain) = 0.7. Find P(no rain)." → 1 − 0.7 = 0.3.
- "A die is biased so P(6) = 1/2 and other outcomes equally likely. Find P(1)." → Other 5 outcomes share 1 − 1/2 = 1/2 → each is 1/10.
Decimal vs fraction vs percentage
OCR accepts all three. Use whichever is most natural:
- Probabilities given as percentages should be converted to decimals (or made consistent) before adding/subtracting.
- 0.3 + 25% + 1/4 ≡ 0.3 + 0.25 + 0.25 = 0.8.
Two-event special cases
For mutually exclusive events A and B (cannot both happen at the same trial):
- P(A or B) = PA + PB.
For independent events: P(A and B) = PA × PB. [Higher tier mainly.]
OCR mark scheme conventions
- "Find the probability of …" with sum-to-1 step: M1 for using sum = 1, A1 for value.
- Final answer in any of fraction/decimal/percentage accepted unless restricted.
⚠Common mistakes
- Forgetting that probabilities must sum to 1 — leaving an answer that puts total > 1 or < 1.
- Adding fractions with different denominators incorrectly.
- Computing 100 − P (as if working in percentage) when P is already a decimal.
- Treating "OR" the same as "AND" for non-mutually-exclusive events.
AI-generated · claude-opus-4-7 · v3-ocr-maths-leaves