Problem 107 - A Complete Resource Book in Mathematics for JEE Main

Question

In a certain recruitment test there are multiple choice questions. There are 4 possible answers to each question and of which one is correct. An intelligent student knows \(90 \%\) of the answer while a weak student knows only \(20 \%\). If an intelligent student gets the correct answer, then the probability that he was guessing is (A) \(\frac{1}{37}\) (B) \(\frac{36}{37}\) (C) \(\frac{14}{37}\) (D) none of these

Step-by-Step Solution

Answer

(A) \( \frac{1}{37} \).

Step 1: Understand the Problem Statement

We need to find the probability that an intelligent student was guessing when they answered a question correctly. The intelligent student knows 90% of the answers and there are 4 options for each question, so if they don't know the answer, they guess among the remaining 3 options.

Step 2: Apply Conditional Probability

We're looking for the conditional probability that a student was guessing, given that they answered correctly. We use the formula: \[ P( ext{Guessing | Correct}) = \frac{P( ext{Correct | Guessing}) \, P( ext{Guessing})}{P( ext{Correct})} \] where \( P(\text{Correct | Guessing}) = \frac{1}{4} \), since the student is guessing among 4 options.

Step 3: Calculate Probabilities

First, calculate \( P(\text{Correct}) \). It can occur if:- The student knows the answer, with probability 0.9, or- The student guesses correctly, which is \( 0.1 \times \frac{1}{4} = 0.025 \).Thus, \( P(\text{Correct}) = 0.9 + 0.025 = 0.925\).

Step 4: Plug Values into Conditional Probability Formula

Now substitute the values into the conditional probability formula: \[ P(\text{Guessing | Correct}) = \frac{\frac{1}{4} \times 0.1}{0.925} \]\[ = \frac{0.025}{0.925} \]

Step 5: Simplify the Expression

Simplify \( \frac{0.025}{0.925} \) using division:\[ \frac{0.025}{0.925} = \frac{25}{925} \]Divide both numerator and denominator by 25:\[ \frac{25}{925} = \frac{1}{37} \]

Step 6: Conclusion

Thus, the probability that an intelligent student was guessing when they answered correctly is \( \frac{1}{37} \).

Key Concepts

Probability Theory

Probability Theory is an essential part of mathematics that helps us understand and manage uncertainty. In simple terms, it is the study of how likely an event is to occur.
Let's break down some key terms:

• **Event:** A specific outcome or occurrence. For example, drawing a red card from a deck or getting a correct answer on a test.
• **Sample Space:** All possible outcomes of an experiment. For a single dice throw, it's 1, 2, 3, 4, 5, or 6.
• **Probability:** The measure of the likelihood that an event will occur. It's a value between 0 and 1, where 0 means it will never happen, and 1 means it will definitely happen.

Conditional probability is another important aspect of probability theory.
It's the probability that an event occurs given that another event has already occurred. For example, the probability that a card drawn is a heart, given that it's red, is a conditional probability. In our recruitment test example, we wanted to know the probability the student guessed, given the fact they answered correctly.

Recruitment Test

Recruitment Tests often involve multiple-choice questions designed to assess the knowledge or skills of potential employees. These tests are intended to evaluate how well candidates understand the material and solve problems.
There can be two types of students taking these tests: intelligent students who know most of the answers, and weaker students who know fewer.
Whether they know the answer or not influences how they tackle a question:

• **Knowledgeable Students:** These students are likely to answer correctly due to their understanding of the material (e.g., they know 90% of the answers).
• **Weaker Students:** These students might have to rely on guessing more often (e.g., they know only 20% of the answers).

Understanding the strategies that students use, whether guessing or relying on knowledge, can help in evaluating their performance accurately or interpreting the probability of certain outcomes in tests.

Multiple Choice Questions

Multiple Choice Questions (MCQs) are a common format for tests, particularly recruitment tests. Each question presents a prompt with several possible answers, usually one correct answer and several distractors.
The student has to select the best answer from the given options. Here are a few details about MCQs:

• **Random Guessing:** If a student guesses, they have a one in four chance of getting the correct answer, assuming four choices.
• **Probability and Strategy:** For students who don't know all the answers, guessing might increase their chances of answering correctly, especially under time pressure.
• **Assessment Accuracy:** Analyzing test patterns, such as the number of correct answers resulting from guessing, can provide insight into the reliability of the test in measuring true knowledge.

In the context of our problem, understanding how often an intelligent student might resort to guessing and still get the answer correct plays into calculating conditional probabilities. By knowing these factors, assessments can be structured to capture not only what a candidate knows, but also how they apply their understanding to unfamiliar questions.