Which statement about conditional probability is true?

Which statement about conditional probability is true?

• A. P(A|B) = P(A) * P(B)
• B. P(A|B) = P(A and B) / P(B)
• C. P(A|B) = P(A) + P(B) - P(A and B)
• D. P(A|B) = P(B|A)

Answer: (B)

Conditional probability measures how likely A is when we know B has occurred. It is defined as P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0. This reflects focusing on the part of the sample space where B happens and asking what fraction of that part also has A. So the true statement is that P(A|B) equals P(A ∩ B) divided by P(B). Keep in mind the other forms aren’t generally true. P(A|B) = P(A) × P(B) would only hold if A and B are independent. The expression P(A|B) = P(A) + P(B) − P(A ∩ B) is the formula for the probability of A or B, not conditional probability. And P(A|B) = P(B|A) is not generally true; it would require a special relationship between A and B. Example: if P(B) = 0.5 and P(A ∩ B) = 0.2, then P(A|B) = 0.2 / 0.5 = 0.4, illustrating how conditioning on B changes the sample space to B and measures A within that space.

Conditional probability measures how likely A is when we know B has occurred. It is defined as P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0. This reflects focusing on the part of the sample space where B happens and asking what fraction of that part also has A. So the true statement is that P(A|B) equals P(A ∩ B) divided by P(B).

Keep in mind the other forms aren’t generally true. P(A|B) = P(A) × P(B) would only hold if A and B are independent. The expression P(A|B) = P(A) + P(B) − P(A ∩ B) is the formula for the probability of A or B, not conditional probability. And P(A|B) = P(B|A) is not generally true; it would require a special relationship between A and B.

Example: if P(B) = 0.5 and P(A ∩ B) = 0.2, then P(A|B) = 0.2 / 0.5 = 0.4, illustrating how conditioning on B changes the sample space to B and measures A within that space.