Problem 18 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

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) \(\frac{1}{4}\)

Step-by-Step Solution

Answer

The probability that only two tests are needed is \(\frac{1}{3}\).

Step 1: Understand the scenario

There are 4 machines in total, out of which exactly 2 machines are faulty. We test these machines one by one until both faulty machines are found. We need to determine the probability that only 2 tests are needed to find both faulty machines.

Step 2: Determine the initial condition

For exactly 2 tests to find both faulty machines, the first test must identify a faulty machine, and the second test must also identify another faulty machine. This means both tests must be conducted on the faulty machines.

Step 3: Calculate combinations

First, we count the number of ways to choose 2 faulty machines from the 4. This is given by the combination formula \(^nC_r = \frac{n!}{r!(n-r)!}\). For our problem, the number of ways to choose 2 faulty machines from 4 is \(^4C_2 = \frac{4!}{2!2!} = 6\).

Step 4: Calculate successful outcomes

The successful outcome is choosing the first faulty machine in the first test and the second faulty machine in the second test. After choosing any of the two faulty machines for the first test (2 ways), the second test must be the other faulty machine (1 way). Hence, there are 2 successful outcomes where both tests identify faulty machines.

Step 5: Compute the total probability

The probability is calculated as the number of successful outcomes divided by the total number of ways to select two machines (for both tests) from the four machines. Probability = \( \frac{2}{6} = \frac{1}{3} \).

Step 6: Identify the correct answer

Compare the calculated probability \(\frac{1}{3}\) to the given options: (A) \(\frac{1}{3}\), (B) \(\frac{1}{6}\), (C) \(\frac{1}{2}\), and (D) \(\frac{1}{4}\). The correct answer is (A) \(\frac{1}{3}\).

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arranging, and finding patterns in sets of objects. Imagine you have a box of different colored marbles and you want to know how many ways you can choose two marbles from this box. This is where combinatorics comes in handy. It provides us tools and formulas to calculate such possibilities.
For example, one of the key formulas in combinatorics is the combination formula. It's used to calculate the number of ways you can choose a certain number of objects from a larger group, without considering the order.
The formula is: \[ ^nC_r = \frac{n!}{r!(n-r)!} \]where:

• n is the total number of objects,
• r is the number of objects you want to choose,
• ! denotes factorial, which is the product of an integer and all the integers below it (e.g., 4! = 4 × 3 × 2 × 1 = 24).

Applying this formula helps solve problems like selecting faulty machines or any instance where choices need counting without regard for sequence.

Faulty Machines

In our problem, we are dealing with four machines, two of which are faulty. Faulty machines in this context are machines that do not function as expected. Identifying them quickly is crucial, especially in manufacturing or production settings where efficiency is key.
When you know that exactly two machines are faulty out of a set group, you can use strategic testing to figure out the faulty ones with the least amount of trials. The goal is to minimize operational disruptions.
Interestingly, in the context of the problem, finding two faulty machines quickly means you need to identify these machines in just two tests. This requires an understanding of probability and careful planning of test sequences. Emphasizing the importance of sequence in testing, the first test needs to identify a faulty machine and so does the second test to meet the condition of finding both faulty machines in only two attempts.

Probability Calculation

Probability calculation involves determining the likelihood of a particular event happening. In the case of our faulty machines, this means calculating how likely it is to find both faulty machines in just two tests. When you calculate probability, you compare successful outcomes with the total possible outcomes. Using our exercise, the probability that exactly two tests are needed involves:

• First, the successful outcome: You need to select two faulty machines on two tests, which can happen in 2 ways (one for each of the faulty machine pairs).
• Next, understand the total outcomes: Choosing 2 machines in sequence from a total of 4 gives you 6 different outcomes (without regard to which are faulty or not).

Thus, the calculation is: \[ \text{Probability} = \frac{\text{Number of Successful Outcomes}}{\text{Total Possible Outcomes}} = \frac{2}{6} = \frac{1}{3} \]. This probability tells us there's a one in three chance, under random testing, that the first two tests will both find the faulty machines.