Problem 107 - 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

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

Step 1: Understanding the Problem

We are given probabilities for an intelligent student's ability to know correct answers and the problem asks for the probability that the student was guessing given they answered correctly. This involves conditional probability.

Step 2: Define the Probabilities

Let A be the event that the student was guessing. Let B be the event that the student correctly answers the question. We need to find \(P(A|B)\), the probability that the student was guessing given they got the answer right.

Step 3: Calculate the Probability of Different Events

Since the intelligent student knows 90% of the answers, the probability that they need to guess is \(P(A) = 0.10\). The probability of a correct answer if guessing is \(P(B|A) = \frac{1}{4}\), since there's only one correct answer out of four choices.

Step 4: Total Probability of Correct Answer

Calculate the total probability that the student answers correctly, \(P(B)\). This can happen if they know the answer or if they guess correctly. Thus, \(P(B) = 0.9 + 0.1 \times \frac{1}{4} = 0.9 + 0.025 = 0.925\).

Step 5: Apply Conditional Probability Formula

Use the formula for conditional probability: \(P(A|B) = \frac{P(A \cap B)}{P(B)}\), where \(P(A \cap B) = P(A) \times P(B|A) = 0.10 \times 0.25 = 0.025\). This gives us \(P(A|B) = \frac{0.025}{0.925}\).

Step 6: Simplify the Result

Simplify \(P(A|B) = \frac{0.025}{0.925}\) to get \(\frac{1}{37}\). After doing the calculation, this fraction represents the probability that the intelligent student guessed the answer given that they got it correct.

Key Concepts

Bayes' Theorem

Understanding Bayes' Theorem is crucial when analyzing situations involving conditional probabilities. It helps us reverse conditional dependencies, allowing us to find the probability of an initial hypothesis based on observed outcomes. In the context of the exercise, Bayes' Theorem enables us to calculate the probability that an intelligent student was guessing, given that they answered correctly.

Bayes' Theorem is expressed as:\[P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}\]Where:

• \( P(A|B) \) is the probability that hypothesis \( A \) is true given event \( B \) has occurred.
• \( P(B|A) \) is the probability of observing event \( B \) assuming hypothesis \( A \) is true.
• \( P(A) \) is the prior probability of hypothesis \( A \).
• \( P(B) \) is the total probability of observing event \( B \).

By applying these elements to the problem:
- \( P(A) = 0.10 \) is the probability the student guesses.
- \( P(B|A) = \frac{1}{4} \) is the probability of a correct guess.
- \( P(B) = 0.925 \) is the probability of answering correctly, whether by knowing or guessing.
Substitute these values into Bayes' formula to find the desired conditional probability.

Probability Theory

Probability theory is a fundamental mathematical framework used to measure the likelihood of outcomes. It is pivotal in understanding and calculating events in a wide array of scenarios, from simple games to complex experiments. Mathematically, it is used to predict, analyze, and interpret the behavior of systems involving randomness.

In our exercise, probability theory helps us break down events such as:

• The probability of the student guessing the correct answer.
• The probability of the student knowing the correct answer.
• The combined probability of getting an answer correct whether by knowledge or guessing.

Each of these probabilities aids in building a more comprehensive understanding of the situation at hand.

For instance, the probability of knowing the answer \( P(K) = 0.9 \) and the probability of guessing correctly \( P(G|C) = \frac{1}{4} \) because there are four options.
Understanding these parts and how they interact according to the rules of probability allows us to perform calculations such as determining the total probability of accurately answering questions on the test.

Problem Solving in Mathematics

Problem solving is a skill crucial not only in mathematics but also in day-to-day decision-making. Within mathematics, it involves identifying the underlying principles, structuring the problem in terms of known concepts, and applying calculations appropriately.

To tackle a problem like the one given, the correct approach often involves:

• Decomposing the problem into recognizable events and assigning relevant probabilities.
• Understanding the meaning of terms like conditional probability and its calculation procedure.
• Applying formulas, such as Bayes' Theorem, accurately and simplifying results.

Step-by-step solutions, such as defining events \( A \) (guessing) and \( B \) (correct answer), help break down the task.
Properly executing these steps allows for solving the problem logically, leading to reliable results. As demonstrated, correctly interpreting results also apples simplicity to the apparent complexity of mathematical exercises.