Problem 47 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

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

Step-by-Step Solution

Answer

(a) \(\frac{3}{4}\)

Step 1: Calculate the Probability of Not Solving the Problem

To find the probability of the problem being solved by at least one student, we first calculate the probability that no student solves the problem. If the probability of solving the problem for students A, B, and C are \(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}\) respectively, then the probabilities of them not solving it are \(1 - \frac{1}{2} = \frac{1}{2}\), \(1 - \frac{1}{3} = \frac{2}{3}\), and \(1 - \frac{1}{4} = \frac{3}{4}\).

Step 2: Compute the Probability That None Solve the Problem

We multiply the probabilities that each student does not solve the problem to find the probability that none of them solve it: \[ P(A' \cap B' \cap C') = \frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} = \frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} = \frac{1}{4} \]

Step 3: Find the Probability That the Problem is Solved

The complementary probability (the probability that at least one of the students solves the problem) is calculated by subtracting the probability of none solving it from 1. Thus, \[ P(A \cup B \cup C) = 1 - P(A' \cap B' \cap C') = 1 - \frac{1}{4} = \frac{3}{4} \]

Key Concepts

Complementary Events

In probability, complementary events are two outcomes of an event that are mutually exclusive and together capture all possible outcomes. For example, if you consider the event of flipping a coin, the complementary events are getting heads or getting tails. They cannot occur simultaneously, and one of them will happen every time a coin is flipped.

In the mathematical problem given, we are dealing with the event of the problem being solved. Its complementary event is the problem not being solved. To find the probability of the problem being solved by at least one student, we start by calculating the complementary event's probability, which is that none of the students solves the problem:

• The probability of student A not solving the problem is \(1 - \frac{1}{2} = \frac{1}{2}\).
• For student B, it is \(1 - \frac{1}{3} = \frac{2}{3}\).
• For student C, it is \(1 - \frac{1}{4} = \frac{3}{4}\).

To compute the probability that none of the students solves the problem, the complementary probabilities are multiplied as they are independent events. This complements the probability of at least one solving it, which is central to finding whether the task will be accomplished by any of them.

Independent Events

Independent events in probability are events whose occurrence does not affect the probability of each other happening. In the context of our math problem, whether one student solves the problem does not influence the ability or likelihood of the other students solving it.

This is a perfect scenario for discussing independent events since:

• The probability of student A not solving the problem is \(\frac{1}{2}\), independent of the others.
• Student B's probability of not solving it is \(\frac{2}{3}\), also unaffected by others.
• Student C's probability of not solving it remains \(\frac{3}{4}\).

When calculating that none of them solves the problem, you can multiply these probabilities because they are independent events. This gives you a composite probability of none solving the problem, which is crucial when you want to compute the likelihood of at least one student solving it.

By understanding independence, we apply the principle that the joint probability of all individual events occurring is the product of their individual probabilities. This principle allows us to determine complementary outcomes that are so useful in problem-solving.

Mathematics Problem Solving

Mathematics problem-solving often involves breaking down a complex question into simpler parts and using known theories or rules to find the solution. In probability, this typically means understanding different possible outcomes, calculating their likelihood, and using logical reasoning to combine them.

In the student problem-solving scenario, we followed these steps:

• Identify the individual probabilities of solving the problem for each student.
• Calculate the complementary probabilities, which is the probability of not solving the problem for each student.
• Use multiplication rules for independent events to find the probability that none solve the problem.
• Apply the concept of complementary events to find the probability that at least one solves it by subtracting the complement's probability from one.

Engaging in these steps requires not only an understanding of probability but also strategic thinking and logical progression to navigate through mathematical exercises effectively. By skillfully using concepts like complementary and independent events, we solve seemingly complex problems with clarity and correctness.