Which statement correctly describes the marginal probability P(A) in a joint distribution with B?

Which statement correctly describes the marginal probability P(A) in a joint distribution with B?

• A. P(A) equals sum over B of P(A,B)
• B. P(A) equals P(A|B)
• C. P(A) equals P(B|A)
• D. P(A) equals P(A,B) for a single B

Answer: (D)

When two variables are described together, the marginal probability of the first one is what you get when you combine all the ways the second variable can take on any value and add them up. In discrete settings, P(A) is found by summing the joint probability across all possible values of B: P(A) = sum over all b of P(A, B = b). This removes any dependence on B and tells you the overall chance that A occurs, regardless of B. If B were continuous, you’d instead integrate: P(A) = ∫ P(A, B) dB. This differs from P(A|B), which fixes a value of B and asks how likely A is given that specific B. To recover P(A) from conditional probabilities, you’d average over B: P(A) = sum_b P(A|B=b) P(B=b) (or the analogous integral for continuous B). It also differs from P(B|A), which is the distribution of B given A. And P(A,B) for a single value of B is the joint probability of A and that particular B, not the overall marginal probability of A across all B.

When two variables are described together, the marginal probability of the first one is what you get when you combine all the ways the second variable can take on any value and add them up. In discrete settings, P(A) is found by summing the joint probability across all possible values of B: P(A) = sum over all b of P(A, B = b). This removes any dependence on B and tells you the overall chance that A occurs, regardless of B.

If B were continuous, you’d instead integrate: P(A) = ∫ P(A, B) dB.

This differs from P(A|B), which fixes a value of B and asks how likely A is given that specific B. To recover P(A) from conditional probabilities, you’d average over B: P(A) = sum_b P(A|B=b) P(B=b) (or the analogous integral for continuous B). It also differs from P(B|A), which is the distribution of B given A. And P(A,B) for a single value of B is the joint probability of A and that particular B, not the overall marginal probability of A across all B.