Introduction to Probability Theory
Introduction to Probability Theory sets up the language of random experiments and events, then applies the addition theorem, conditional probability, the multiplication theorem and independence. Most marks come from carefully counting the sample space.
Rules and axioms
For a sample space S and event A, 0 ≤ P(A) ≤ 1, P(S) = 1, P(∅) = 0, and P(A′) = 1 − P(A). With equally likely outcomes, P(A) = n(A)/n(S). The addition theorem states P(A∪B) = P(A) + P(B) − P(A∩B); when the events are mutually exclusive the last term is zero. Conditional probability is P(A|B) = P(A∩B)/P(B) for P(B) ≠ 0, and the multiplication theorem gives P(A∩B) = P(A)P(B|A). Two events are independent when P(A∩B) = P(A)P(B), equivalently P(A|B) = P(A).
Worked example
Two cards are drawn one after another from a well-shuffled pack without replacement. The probability that both are aces is (4/52)(3/51) = 12/2652 = 1/221. The numerator and the denominator both drop by one because the first ace is not replaced.
Traps
- Treating mutually exclusive events as independent; they cannot occur together, so they are not.
- Forgetting to reduce the denominator after a draw without replacement.
- Using P(A∪B) = P(A) + P(B) when the events overlap.
- Counting the sample space without checking whether outcomes are equally likely.