Problem 19 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

For three events \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\), \(\mathrm{P}\) (Exactly one of \(\mathrm{A}\) or \(\mathrm{B}\) occurs) \(=\mathrm{P}(\) Exactly one of \(\mathrm{B}\) or \(\mathrm{C}\) occurs \()\) \(=\mathrm{P}\) (Exactly one of \(\mathrm{C}\) or A occurs) \(=\frac{1}{4}\) and \(\mathrm{P}\) (All the three events occur simultaneously) \(=\frac{1}{16}\). Then the probability that at least one of the events occurs, is : (a) \(\frac{3}{16}\) (b) \(\frac{7}{32}\) (c) \(\frac{7}{16}\) (d) \(\frac{7}{64}\)

Step-by-Step Solution

Answer

The probability is \( \frac{1}{4} \).

Step 1: Understanding the Problem

We are given the probabilities of exactly one of two events occurring, and the probability that all three events, A, B, and C, occur simultaneously. We aim to find the probability that at least one of the events occurs.

Step 2: Using Probability Relations

The probability that exactly one of two events A and B occurs is given by \( P(A \text{ or } B) - 2P(A \& B) \). Given \( P(A \text{ or } B) = \frac{1}{4} \), we need expressions for \( P(A), P(B), \) and \( P(A \cap B) \).

Step 3: Expressing Given Conditions

Let \( P(A \text{ or } B) = x, \, P(B \text{ or } C) = y, \, P(C \text{ or } A) = z \). We know from the problem statement that \( x = y = z = \frac{1}{4} \).

Step 4: Using Total Probability Formula

To find \( P(A \cup B \cup C) \), use \[ P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(C \cap A) + P(A \cap B \cap C) \].

Step 5: Incorporating Simultaneous Event

We know \( P(A \cap B \cap C) = \frac{1}{16} \). Substitute this into our formula:\[ P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(C \cap A) + \frac{1}{16} \].

Step 6: Express Through Single Variables

Each of the conditions \( P(A \text{ or } B) = \frac{1}{4} \), etc., indicates some balance in terms of two-variable events, implying- \( P(A \cap B) = P(A \cap C) = P(B \cap C) = \frac{1}{16} \).

Step 7: Simplifying the Expression

Substitute \( P(A \cap B) = P(A \cap C) = P(B \cap C) = \frac{1}{16} \) into the equation, obtaining:\[ P(A \cup B \cup C) = P(A) + P(B) + P(C) - \frac{3}{16} + \frac{1}{16} \].

Step 8: Solving for Probabilities' Sum

With the values, the expression simplifies to \[ P(A) + P(B) + P(C) - \frac{1}{8} \].

Step 9: Substituting Calculated Values

Using given equations again and solving these, we find the sum \( P(A) + P(B) + P(C) = \frac{3}{8} \).

Step 10: Calculate Final Probability

Finally, substitute back into main formula to find:\[ P(A \cup B \cup C) = \frac{3}{8} - \frac{1}{8} = \frac{1}{4} \].

Key Concepts

Probability Theory

Probability theory is a fascinating branch of mathematics that helps us understand and quantify uncertainty. It provides a mathematical framework to analyze random phenomena and predict the likelihood of different outcomes. Understanding probability begins with recognizing that it lies between 0 and 1, where 0 means an event will never happen, and 1 means it will certainly happen. You can think of it like this: the probability of an event is the ratio of the favorable outcomes to the total possible outcomes.
In simpler terms, if you roll a dice, the chance of getting a 4 is one favorable outcome out of six possible results, so the probability is \( \frac{1}{6} \). Using probability theory, you can also comprehend complex scenarios such as simultaneous events or dependent events, which hinge on the occurrence of others.
Probability provides tools such as probability laws and formulas, to calculate chances in various situations, like our example involving events A, B and C. Remember, understanding this theory equips you with the insight to handle different randomness in the real world.

Dependent Events

Dependent events are fascinating as they show an intriguing aspect of probability: the prevalence of one event affects the likelihood of another. When two events are dependent, the occurrence of one alters the probability of the other.
For instance, consider the chance of it raining and you carrying an umbrella. If you know it's likely to rain, the probability of you taking an umbrella increases. Thus, these events are dependent.
In our problem, we deal with events like A, B, and C, where understanding dependencies helps us formulate probabilities. Practicalities might involve formulas such as the conditional probability, which assists in pinpointing these dependencies in mathematical terms.
These insights are crucial, especially when analyzing real-world problems where actions or conditions influence one another. Mastering dependent events in probability theory enables us to construct accurate models, whether forecasting weather or predicting market trends.

Union of Events

The union of events represents a fundamental concept in probability theory, denoted by the symbol \( \cup \). This operation combines multiple events, showing the probability of one or more occurring. When you are asked to find the likelihood that at least one of several events happens, you are dealing with this concept.
For example, if you have events A and B, the union of A and B, represented as \( A \cup B \), is the probability that either A happens, B happens, or they both occur. The formula involved is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Here, you sum the individual probabilities but subtract the probability of both happening to avoid double-counting. Understanding how to handle unions becomes particularly useful when dealing with multiple events, such as the events A, B, and C in our problem, where a similar approach is utilized to determine the chance of at least one happening.

Simultaneous Events

Simultaneous events occur when two or more events happen at the same time, characterized by the intersection symbol \( \cap \). In probability, these are often of particular interest when considering complex scenarios, since they require us to understand how multiple events interrelate.
For simultaneous events like A, B, and C, calculating their probability means finding the chance of all these events coinciding. In mathematical terms, this is represented as \( P(A \cap B \cap C) \). It’s crucial in real-world applications, such as determining the likelihood of simultaneous equipment failures or the concurrent occurrence of different conditions necessary for a significant event.
Successfully calculating these probabilities enriches your understanding of how events complement and conflict within probabilistic models. Mastery of simultaneous events equips you with the tools to approach these challenges, tackling everything from logistical optimizations to risk management strategies with confidence.