I realize I’m a couple weeks behind and nobody might read this, but this is false (e.g., if I flip two fair + independent coins and A = “first coin is heads” and B = “second coin is heads”, P(B) = 1⁄2, but P(B|A) x P(A) = (1/2)x(1/2) = 1⁄4). I think what you mean is P(A and B) = P(B|A) x P(A).
(unless I’m missing some context and we are assuming B is a subset of A, but if so I’m not sure why we are assuming that)
I realize I’m a couple weeks behind and nobody might read this, but this is false (e.g., if I flip two fair + independent coins and A = “first coin is heads” and B = “second coin is heads”, P(B) = 1⁄2, but P(B|A) x P(A) = (1/2)x(1/2) = 1⁄4). I think what you mean is P(A and B) = P(B|A) x P(A).
(unless I’m missing some context and we are assuming B is a subset of A, but if so I’m not sure why we are assuming that)