Which expression correctly states the law of total probability for a partition B_i of the sample space?

• A. P(A) = sum P(A|B_i) P(B_i) for a partition {B_i}.
• B. P(A) = sum P(B_i|A) P(A).
• C. P(A) = max_i P(A|B_i) P(B_i).
• D. P(A) = P(A ∩ B) + P(A ∪ B).

Answer: (A)

When a set of events B_i partitions the sample space, you can split any event A into the pieces that lie inside each B_i. Those pieces A ∩ B_i are disjoint and together cover A, so you can add their probabilities to get P(A). Using the definition of conditional probability, P(A ∩ B_i) = P(A|B_i) P(B_i). Put together, P(A) = Σ_i P(A|B_i) P(B_i). This is the standard law of total probability: it says the chance of A is the sum of A’s chances within each scenario B_i, weighted by how likely each scenario is.

This form is the most direct and widely used way to relate P(A) to the conditional probabilities given the partition B_i. The other expressions don’t capture the way A is distributed across the different B_i, and one of them relies on a product that isn’t generally how total probability is expressed.