Sum of probabilities = 1
A foundational result Edexcel tests on every Foundation paper. Used to:
- Find a missing probability when others are given.
- Compute a complement (P(not A) = 1 − PA).
- Check that a probability table is valid.
The result
For an experiment with mutually exclusive, exhaustive outcomes A₁, A₂, ..., Aₙ:
P(A₁) + P(A₂) + ... + P(Aₙ) = 1.
Finding a missing probability
Worked example: a biased dice has P(1) = 0.1, P(2) = 0.15, P(3) = 0.2, P(4) = 0.15, P(5) = 0.2. Find P(6). Sum so far: 0.1 + 0.15 + 0.2 + 0.15 + 0.2 = 0.8. P(6) = 1 − 0.8 = 0.2.
Complementary events
P(not A) = 1 − PA.
Example: P(it rains) = 0.35. P(it does not rain) = 1 − 0.35 = 0.65.
With fractions
P(red) = 1/4, P(blue) = 1/3. P(any other colour) = 1 − 1/4 − 1/3. Common denominator 12: 1 − 3/12 − 4/12 = 5/12.
Edexcel exam tip
For "the probabilities are p, q, r, s, find s" questions, always show the equation explicitly: p + q + r + s = 1. The M1 mark requires this. Then rearrange.
Validating probability tables
If the values given don't sum to 1, the table is invalid. Edexcel asks: "Karim claims a probability table — explain why this is wrong" — sum the values; show ≠ 1.
Common Edexcel question pattern
A bag contains red, blue, green, and yellow balls. P(R) = 0.4, PB = 2 × P(G), P(Y) = 0.15. Find PB and P(G).
Solution: Let P(G) = g. Then PB = 2g. 0.4 + 2g + g + 0.15 = 1. 3g = 0.45 ⇒ g = 0.15. So P(G) = 0.15 and PB = 0.30.
⚠Common mistakes— Common errors
- Adding probabilities outside the same experiment (e.g. P from one event + P from another).
- Sum > 1 — flag immediately as invalid.
- Negative probability — impossible.
- Forgetting that "any colour other than red" = 1 − P(red), not the sum of remaining listed.
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