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Rules of Probability

Everything in probability follows from three axioms:

  1. Non-negativity. P(A)≥0P(A) \geq 0 for every event.
  2. Normalization. P(Ω)=1P(\Omega) = 1: something in the sample space must happen.
  3. Additivity. If AA and BB are mutually exclusive, P(A∪B)=P(A)+P(B)P(A \cup B) = P(A) + P(B).

From those, a handful of working rules follow. What matters in practice is less the rules themselves than the condition each one requires, because misapplying a rule whose condition fails is the most common way to get a probability question wrong.

The addition rule

For any two events, whether or not they overlap:

The addition rule
P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)

Add the two, then subtract the overlap you have just counted twice.

The subtraction removes the double count of the overlap. Draw one card from a standard deck and ask for the probability it is a heart or a face card:

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