Directions: Each question contains Statement-1 and...

Directions: Each question contains Statement-1 and Statement-2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct. (a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement- 1 (b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1 (c) Statement- 1 is True, Statement- 2 is False (d) Statement- 1 is False, Statement- 2 is True In a workshop, there are 10 shaping machines of which 3 are known to be defective. In order to identify the defective machines, they are tested one by one. Statement 1 The probability that exactly 6 tests are needed to identify the defectives is $\frac{1}{12}$. and Statement 2 Number of ways of selecting 6 machines out of 10 machines is ${ }^{10} \mathrm{C}_{4}$.

Directions: Each question contains Statement-1 and Statement-2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct.
(a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement- 1
(b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1
(c) Statement- 1 is True, Statement- 2 is False
(d) Statement- 1 is False, Statement- 2 is True
In a workshop, there are 10 shaping machines of which 3 are known to be defective. In order to identify the defective machines, they are tested one by one.
Statement 1
The probability that exactly 6 tests are needed to identify the defectives is $\frac{1}{12}$. and
Statement 2 Number of ways of selecting 6 machines out of 10 machines is ${ }^{10} \mathrm{C}_{4}$.

Key Concepts

Sequential Testing in Quality Control

This concept involves testing items one at a time until a specified condition or set of outcomes is achieved, such as identifying all defective items. It emphasizes not only which items are defective but also the order in which defects are detected, which affects the overall probability calculation in a sequential process.

Counting Arrangements Using Combinatorial Analysis

This principle uses the techniques of combinatorics to count the number of ways events or selections can occur, particularly when order matters or when selections are to be made from a finite set. It provides the tools for calculating the total or favorable outcomes needed to determine probabilities in complex scenarios.

Order-specific Probability Calculation

When outcomes occur in a sequence, the probability of a particular event can depend on its specific position in the sequence. This concept involves understanding and calculating the number of favorable arrangements where an event (such as the detection of the final defective item) occurs at a predetermined test number, taking into account the ordering of successes and failures.

Binomial Coefficient Interpretation

The binomial coefficient, often written as C(n, k), quantifies the number of ways to choose k items from a set of n items without regard to order. This concept is critical in problems dealing with combinations and selections, providing a foundation for calculating probabilities in scenarios where the order of selection is adjusted or when considering subsets within a larger group.

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