Problem 108 - A Complete Resource Book in Mathematics for JEE Main 2017

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

The probability is \( \frac{1}{37} \), so the answer is \( \text{(A)} \).

Step 1: Understanding the Problem

To solve this problem, we need to find the probability that an intelligent student guessed an answer, given that the answer is correct. This can be approached using Bayes' Theorem, involving conditional probabilities.

Step 2: Define Probabilities

Let event A be the event that the intelligent student was guessing the answer. Let event B be the event that the answer given by the student is correct. We know that the probability that an intelligent student knows the answer (not guessing) is 0.9 and thus the probability they were guessing is 0.1.

Step 3: Apply Bayes' Theorem

Bayes' theorem states that \( P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} \). We need to find \( P(A|B) \), the probability the student guessed the answer given that it is correct. \( P(B|A) \), the probability the answer is correct when guessed, is \( \frac{1}{4} \) because there is one correct answer out of four. \( P(A) = 0.1 \).

Step 4: Calculate Total Probability of Correct Answer

The total probability \( P(B) \) that the answer is correct can be calculated by considering both guessed and known probabilities: \( P(B) = P(B|A)P(A) + P(B|eg A)P(eg A) \). Substitute the known values: \( P(B) = \frac{1}{4} \times 0.1 + 1 \times 0.9 = 0.025 + 0.9 = 0.925 \).

Step 5: Calculate the Conditional Probability

Finally, plug these into Bayes' theorem: \( P(A|B) = \frac{0.1 \cdot \frac{1}{4}}{0.925} = \frac{0.025}{0.925} = \frac{1}{37} \).

Key Concepts

Conditional Probability

Understanding conditional probability is crucial when working with problems like those involving Bayes' Theorem. Conditional probability is essentially the likelihood of an event occurring given that another event has already occurred. For instance, in this exercise, we're interested in the probability that a student was guessing (event A), given that they got the answer correct (event B). This is expressed as \(P(A|B)\).

Bayes' Theorem is a helpful tool for these kinds of problems because it provides a way to update the probability of an event based on new evidence. The theorem is expressed as follows:

\[ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} \]

Here:

• \(P(A|B)\) is the conditional probability of A occurring given B has occurred.
• \(P(B|A)\) is the probability of B occurring given A has occurred.
• \(P(A)\) is the initial probability of A.
• \(P(B)\) is the probability of B occurring.

These concepts are fundamental when applying conditional probability to solve complex problems, such as determining how often students guess correctly.

Probability Theory

Probability theory is a branch of mathematics that deals with the likelihood of events occurring. It provides the tools and rules we need to understand and calculate how likely outcomes or events are under various conditions.

When we explore questions involving multiple choice tests, we are primarily interested in calculating the probability of getting a question correct by either knowing the answer or guessing. Probability theory allows us to measure this likelihood and to express it mathematically, helping us to make sense of the world around us.

In the problem, we're using concepts from probability theory to determine whether an intelligent student's correct answer was a result of guessing or genuine knowledge. This involves understanding both simple probabilities (like the chance of knowing or guessing) and compound probabilities (like Bayes' theorem), which deal with combinations of different events.

Multiple Choice Questions

Multiple choice questions introduce interesting dynamics in probability exercises. These questions usually have a fixed number of potential answers, and typically only one of these is correct. In the provided exercise, every question features 4 possible answers.

The task requires assessing the probability of choosing the correct answer by knowing it versus by guessing. For instance, if there are four possible options, and the student is guessing, the probability of guessing correctly is \(\frac{1}{4}\) because any one of the four choices could be correct.

In our scenario, intelligent and weak students have different known probabilities of knowing the right answer. Intelligent students know the answer 90% of the time and therefore have a 10% chance of guessing. For less prepared students, the conditions differ, and they may have a higher likelihood of guessing. Understanding these differences is key to applying probability theory effectively in academic contexts, especially when assessing performance or preparing educational assessments.