There are four machines and it is known that exactly two of...

There are four machines and it is known that exactly two of them are faulty. They are tested one by one, in a random order till both the faulty machines are identifed. Then the probability that only two tests are needed is (A) $\frac{1}{3}$ (B) $\frac{1}{6}$ (C) $\frac{1}{2}$ (D) $\frac{1}{4}$

There are four machines and it is known that exactly two of them are faulty. They are tested one by one, in a random order till both the faulty machines are identifed. Then the probability that only two tests are needed is
(A) $\frac{1}{3}$
(B) $\frac{1}{6}$
(C) $\frac{1}{2}$
(D) $\frac{1}{4}$

Key Concepts

Sequential Testing

Sequential testing involves performing tests one after the other and making decisions at each step based on the outcomes observed. When testing is performed until a certain condition is met, such as identifying all faulty items, the sequence and stopping criteria affect the overall probability of different outcomes. This technique is common in quality control and decision-making processes.

Stopping Rules

Stopping rules define when a sequential process should be terminated based on the observed results up to that point. They are essential in problems where the testing process halts upon reaching a specific condition, such as detecting a required number of faulty items. Stopping rules impact the sample space and influence how probabilities are calculated within sequential experiments.

Probability

Probability is the study of the likelihood of events occurring. It involves quantifying the uncertainties in a situation by assigning numerical values between 0 and 1 to outcomes. In problems involving testing and fault detection, probability helps determine the chances of identifying subjects with certain characteristics under different conditions.

Combinatorial Analysis

Combinatorial analysis studies the counting, arrangement, and selection of elements from a set based on specified rules. It is particularly useful in situations where the order of events or selections matters—such as determining the number of ways to arrange tests—and it forms the mathematical foundation for computing probabilities in discrete settings.

Please give Ace some feedback