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: Test each statement independently against the definition and typical application of Bayes theorem, $$P(A|B) = \frac{P(B|A)P(A)}{P(B)}.$$

Step 2: Statement (i) describes exactly what the formula above does: it lets us compute one conditional probability from another conditional probability and the marginal (prior) probability. This is a true description of the theorem, not an opinion, so (i) holds.

Step 3: For statement (ii), think of a concrete case. If $P(A) = 0.5$ before evidence and updating with $B$ gives $P(A|B) = 0.6$, the prior $0.5$ is not erased or overridden - it is simply combined with the likelihood of the evidence to produce the posterior. Since the word "supersedes" implies the prior becomes invalid or is replaced, this description is inaccurate. Statement (ii) is false.

Step 4: For statement (iii), Bayes theorem is the standard tool in problems where a later observation (an effect) is used to infer the probability of an earlier, unobserved cause - such as inferring which machine produced a defective item after observing the defect. This backward-in-time inference is exactly what (iii) describes, so it is true.

Step 5: Only statements (i) and (iii) survive this check.

\[\boxed{\text{Only (i) and (iii) are true}}\]