In a joint distribution of events A and B, the marginal probability P(A) is obtained by:

• A. P(A,B) summed over all B
• B. P(A|B)
• C. P(B) - P(A,B)
• D. P(A,B) for a single B

Answer: (A)

To obtain the marginal probability of A from a joint distribution with B, you collapse the B dimension by summing the joint probabilities over all possible values of B. In discrete terms, P(A) = sum over all B of P(A,B); equivalently, you’re adding up how likely A occurs with every possible value of B. If B were a continuous variable, you would replace the sum with an integral: P(A) = ∫ P(A,B) dB. This is the way to remove B and focus only on A.

The idea behind this is that the joint probability P(A,B) accounts for all ways A can occur together with B. By adding across all B, you collect all those scenarios where A happens, regardless of B.

In contrast, P(A|B) is the probability of A given a specific B, so it does not aggregate over B. P(B) − P(A,B) does not yield a standard marginal and isn’t the method to get P(A). P(A,B) for a single B captures the chance of A occurring with one fixed value of B, not the total chance of A across all B.