Question

The statements below relate to Bayes theorem in probability:

(i) Bayes theorem gives a formula to compute conditional probability.
(ii) The posterior probability computed by Bayes theorem supersedes the prior probability.
(iii) Bayes theorem can be used to compute probabilities of past events on the basis of the occurrences of subsequent events.

Identify the correct answer:

• A. All these statements are true.
• B. Only (i) and (ii) are true.
• C. Only (i) and (iii) are true.
• D. Only (ii) and (iii) are true.

Hint:

Recall that Bayes theorem is a formula for conditional probability, and it is typically used to infer probabilities of earlier causes from later observed effects.

Answer 1

The Correct Option is C

Step 1: Recall what Bayes theorem states. For events \(A\) and \(B\) with \(P(B) > 0\), \[P(A|B) = \frac{P(B|A)P(A)}{P(B)}\] This is a formula that converts a conditional probability given one direction (\(B\) given \(A\)) into the conditional probability in the reverse direction (\(A\) given \(B\)). So statement (i) is true, since Bayes theorem is precisely a rule for computing a conditional probability from other known conditional and marginal probabilities.

Step 2: Check statement (ii). The posterior probability \(P(A|B)\) is an updated probability of \(A\) once the evidence \(B\) has been observed. It does not "supersede" the prior \(P(A)\) in any absolute sense; the prior remains valid before the evidence is known, and the posterior is only the revised belief conditional on that evidence. Calling the posterior a replacement or override of the prior is not correct usage, so statement (ii) is false.

Step 3: Check statement (iii). Bayes theorem is commonly used to reason backward: given that some later event (evidence) has occurred, we compute the probability of an earlier event (a cause) that could have produced it. For example, given a positive test result (a subsequent event), Bayes theorem gives the probability that the disease (an earlier condition) is actually present. So statement (iii) is true.

Step 4: Combining the results, statements (i) and (iii) are true while (ii) is false. This matches option (C).

The correct answer is Only (i) and (iii) are true.