Probability

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The likelihood of an event, expressed as a number between 0 (impossible) and 1 (certain).

For a finite sample space of equally likely outcomes, the probability of an event E is the ratio of favorable outcomes to total outcomes: P(E) = |E| / |S|. Probabilities satisfy Kolmogorov's axioms: 0 ≤ P(E) ≤ 1, P(sample space) = 1, and for mutually exclusive events P(A ∪ B) = P(A) + P(B). For independent events, P(A and B) = P(A) · P(B). Complement rule: P(not E) = 1 − P(E).

P(E) = favorable outcomes / total outcomes; 0 ≤ P(E) ≤ 1

• Rolling a 6 on a fair die → P = 1/6
• Drawing an ace from a 52-card deck → P = 4/52 = 1/13
• Two independent coin flips both heads → P = 1/2 · 1/2 = 1/4

How to recognize it

• The problem asks 'what is the chance' or 'what is the probability'
• A sample space of equally likely outcomes is described (dice, cards, balls in an urn)
• Compound events combine with 'and' (multiply if independent) or 'or' (add if disjoint)

Common mistakes

• Adding probabilities of events that are not mutually exclusive without subtracting overlap
• Multiplying probabilities of events that are not actually independent
• Forgetting the complement rule when the 'at least one' form is easier

Practice guidance

Numerical Reasoning