Probability models for random events — A/L Combined Mathematics

What must I remember about Probability models for random events?

• Classical (Equally Likely) Probability Definition: If sample space S has n equally likely outcomes, and event A contains m outcomes, then P(A) = m / n = n(A) / n(S).
• Axioms of Probability (Kolmogorov): 1) Non-negativity: 0 ≤ P(A) ≤ 1 for every event A; 2) Certainty: P(S) = 1; 3) Additivity: If A and B are mutually exclusive (A ∩ B = ∅), then P(A ∪ B) = P(A) + P(B).
• Complement Rule: P(A') = 1 - P(A); probability of impossible event P(∅) = 0.
• General Addition Rule: For any two arbitrary events A and B: P(A ∪ B) = P(A) + P(B) - P(A ∩ B); for three events: P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A∩B) - P(B∩C) - P(C∩A) + P(A∩B∩C).
• Independent Events definition: Events A and B are Independent if and only if P(A ∩ B) = P(A) * P(B); the occurrence of one does not affect probability of the other.

Test yourself (1 minute)

1. Mutually Exclusive vs Independent distinction

   Mutually exclusive means A and B CANNOT happen together (A ∩ B = ∅ => P(A ∩ B) = 0); Independent means P(A ∩ B) = P(A)*P(B) > 0 for non-zero events; two events with non-zero probability CANNOT be both mutually exclusive and independent!

2. Probability of at least one occurrence

   For independent events A1, A2, ..., An: P(at least one occurs) = 1 - P(none occurs) = 1 - [P(A1') * P(A2') * ... * P(An')].

3. Exactly one of A or B occurs

   P(exactly one) = P(A ∩ B') + P(A' ∩ B) = P(A ∪ B) - P(A ∩ B) = P(A) + P(B) - 2P(A ∩ B).

Practise Probability models for random events questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Applied Mathematics, Unit 7

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Where does Probability models for random events fit in the syllabus?

Year 2 · Term 3