Problem 146 - A Complete Resource Book in Mathematics for JEE Main

Question

A problem in mathematics is given to three students \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and their respective probability of solving the problem is \(\frac{1}{2}, \frac{1}{3}\) and \(\frac{1}{4}\). \([2002]\) Probability that the problem is solved, is: (A) \(3 / 4\) (B) \(1 / 2\) (C) \(2 / 3\) (D) \(1 / 3\)

Step-by-Step Solution

Answer

The probability that the problem is solved is \(\frac{3}{4}\), which corresponds to option (A).

Step 1: Understanding the Event

Define the event that the problem is solved by at least one student. If any of the students solve the problem, then the problem is considered solved.

Step 2: Solving by Complementary Probability

It's easier to find the probability that none of the students solve the problem and then subtract this from 1 to find the probability that at least one student solves it.

Step 3: Calculate Probability of Failure for Each Student

Calculate the probability that student A does not solve the problem as \(1 - \frac{1}{2} = \frac{1}{2}\), student B as \(1 - \frac{1}{3} = \frac{2}{3}\), and student C as \(1 - \frac{1}{4} = \frac{3}{4}\).

Step 4: Probability of None Solving the Problem

Calculate the probability that none of the students solve the problem by multiplying their individual probabilities of failure: \[\frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} = \frac{1 \times 2 \times 3}{2 \times 3 \times 4} = \frac{1}{4}.\]

Step 5: Calculate Probability of At Least One Solving it

Subtract the probability that none of them solve the problem from 1 to find the probability that at least one does: \[1 - \frac{1}{4} = \frac{3}{4}.\]

Step 6: Conclusion

Therefore, the probability that the problem is solved by at least one student is \(\frac{3}{4}\).

Key Concepts

Complementary Probability

Complementary probability is a powerful concept in probability theory that simplifies solving problems where you need to find the likelihood of at least one event occurring. Instead of calculating the probability of each event directly, you find the probability of the complementary event not happening. In our exercise, the complementary event is none of the students solving the problem.

Once you have the probability of none solving the problem, you subtract it from 1 to get the probability of at least one student solving it. This approach is often simpler and reduces the complexity of calculations.

• Find the probability of the unwanted event
• Subtract it from 1 to get the probability of the desired event

This technique is particularly useful when dealing with multiple independent events, which brings us to our next topic.

Probability of Independent Events

Understanding the concept of independent events is critical to solving many probability problems. Independent events are events that do not affect each other's outcomes. In the problem given, each student's ability to solve the problem does not depend on the others. This means their probabilities are independent.

When dealing with independent events, the probability of all events occurring together is the product of their individual probabilities. For our case, we calculated the probability of none solving the problem by multiplying the probabilities of each student not solving it:

• Probability of A not solving: \( \frac{1}{2} \)
• Probability of B not solving: \( \frac{2}{3} \)
• Probability of C not solving: \( \frac{3}{4} \)

The product \( \frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \) gives us \( \frac{1}{4} \). This calculation demonstrates how independent probabilities multiply.

Basic Mathematical Problem Solving

Problem solving in mathematics often requires breaking down complex challenges into manageable steps. By systematically approaching the problem, you ensure clarity and accuracy in finding the solution. In the exercise, the problem was decomposed into clear, simpler tasks, each focusing on a specific aspect;

1. Understand the goal: Identify that the goal is to find the probability that at least one student solves the problem.
2. Use complementary probability: Rather than directly computing the probability of at least one solving it, find the probability of none solving it and then take the complement.
3. Multiply independent probabilities: Calculate the probability of all students failing to solve by multiplying their individual probabilities of failure.

Following these logical steps not only simplifies solving the problem but also builds a strong foundation in mathematical critical thinking. It enhances your ability to handle more complex scenarios in probability and beyond.