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 identified. Then the probability that only two tests are needed is [1998-2 Marks] (a) $1 / 3$ (b) $1 / 6$ (c) $1 / 2$ (d) $1 / 4$

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Key Concepts

Combinatorial Analysis

This concept involves counting the number of possible outcomes (such as arrangements or selections) that satisfy certain conditions. In the context of the problem, it is used to determine the total number of possible positions for the faulty machines among all machines, and the number of favorable outcomes where the faulty machines appear in specific positions.

Probability in Sequential Experiments

This refers to calculating the likelihood of events in a process where experiments or tests are performed one after another until a stopping condition is met. Here, understanding how sequential testing impacts the count of observed outcomes is crucial, as it determines the probability of detecting both faulty machines within a specified number of tests.

Random Orderings

This concept deals with the idea that items (or machines in this case) are arranged in a random order, ensuring that every possible permutation is equally likely. It is essential for the analysis since the order in which the machines are tested affects the probability distribution over the outcomes.

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