A way to get an overall probability by splitting into cases: if B₁, …, Bₙ partition the possible outcomes, P(A) = Σ P(A | Bᵢ) P(Bᵢ). The overall probability is a weighted average of the conditional ones, weighted by how common each case is.
Card 1 of 276. Answer: A way to get an overall probability by splitting into cases: if B₁, …, Bₙ partition the possible outcomes, P(A) = Σ P(A | Bᵢ) P(Bᵢ). The overall probability is a weighted average of the conditional ones, weighted by how common each case is.
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The cases must be mutually exclusive and together cover every outcome, like device type, country or customer segment. Each case contributes its conditional probability of A times its own probability. The same idea for averages is the law of total expectation, E[X] = E[E[X | Y]], and the law supplies the denominator of Bayes' theorem.
Example: 70% of traffic is mobile, converting at 2%, and 30% is desktop, converting at 5%. The overall conversion rate is 0.7 × 0.02 + 0.3 × 0.05 = 2.9%. If the mix shifts to half and half, the overall rate rises to 0.5 × 0.02 + 0.5 × 0.05 = 3.5% although neither segment converted any better.
That mix effect is why interviewers bring this law up: a headline metric can move because the population changed, not because anything improved, and when the mix differs between groups it can reverse a comparison, which is Simpson's paradox. It is also the standard first step in probability puzzles: split on whatever you would like to know, then add up the cases.
