Problem 45 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(A\) and \(B\) be two events such that \(P(\overline{A \cup B})=\frac{1}{6}\), \(P(A \cap B)=\frac{1}{4}\) and \(P(\bar{A})=\frac{1}{4}\), where \(\bar{A}\) stands for complement of event \(A\). Then events \(A\) and \(B\) are [2005] (a) equally likely and mutually exclusive (b) equally likely but not independent (c) independent but not equally likely (d) mutually exclusive and independent

Step-by-Step Solution

Answer

Events \(A\) and \(B\) are independent but not equally likely.

Step 1: Identify Relevant Formulas

To solve this problem, we need three probability concepts: complement rules, probability addition rules, and independence. We know if events are independent, then \( P(A \cap B) = P(A) \cdot P(B) \). Also, we have \( P(\bar{A}) = 1 - P(A) \) and \( P(\overline{A \cup B}) = 1 - P(A \cup B) \).

Step 2: Calculate \(P(A)\)

Given \(P(\bar{A}) = \frac{1}{4}\), use \(P(A) = 1 - P(\bar{A})\) to find \(P(A)\). So, \(P(A) = 1 - \frac{1}{4} = \frac{3}{4}\).

Step 3: Calculate \(P(A \cup B)\) Using Complement

We know \(P(\overline{A \cup B}) = \frac{1}{6}\), therefore, \(P(A \cup B) = 1 - P(\overline{A \cup B}) = 1 - \frac{1}{6} = \frac{5}{6}\).

Step 4: Use Probability Addition Formula

The formula for the union of two events is \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). Plug in the known values: \(\frac{5}{6} = \frac{3}{4} + P(B) - \frac{1}{4}\).

Step 5: Solve for \(P(B)\)

Rearrange the above equation to solve for \(P(B)\): \(P(B) = \frac{5}{6} - \frac{2}{4}\). Simplify the right side to get \(P(B) = \frac{1}{3}\).

Step 6: Check Independence

For two events to be independent, \(P(A \cap B) = P(A) \cdot P(B)\). Check if \(\frac{1}{4} = \frac{3}{4} \cdot \frac{1}{3}\). The right side simplifies to \(\frac{3}{12} = \frac{1}{4}\), so they are independent.

Step 7: Check if Events are Equally Likely

Two events are equally likely if \(P(A) = P(B)\). Since \(P(A) = \frac{3}{4}\) and \(P(B) = \frac{1}{3}\), they are not equally likely.

Key Concepts

Probability principles

Probability is a measure of the likelihood that a particular event will occur. It relies on fundamental principles that help understand random events and their associated uncertainties. There are a few key principles to keep in mind:

• Complement Rule: For any event, the probability that it does not occur is called the complement of the event. If the probability that event \(A\) occurs is \(P(A)\), the probability that \(A\) does not occur, denoted as \(\bar{A}\), is \(1 - P(A)\).
• Addition Rule: This is used to determine the probability of the union of two events (\(A \cup B\)). For any two events \(A\) and \(B\), the probability that at least one of them occurs is \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\).
• Multiplication Rule: This principle helps determine the probability that two independent events occur together. If events \(A\) and \(B\) are independent, their joint probability is \(P(A \cap B) = P(A) \cdot P(B)\).

Each of these principles provides a tool for calculating different aspects of probability, helping us solve various problems, like determining event likelihoods and dependencies.

Event independence

In probability theory, two events are considered independent if the occurrence of one event does not affect the probability of the other. This concept is critical when analyzing event relationships.

• If events \(A\) and \(B\) are independent, it means \(P(A \cap B) = P(A) \cdot P(B)\).
• Independence implies that events do not influence each other.
• Checking independence often involves verifying this mathematical condition using given probabilities.

In practical terms, if you have determined that \(P(A \cap B) = \frac{1}{4}\) and then calculate \(P(A) \cdot P(B) = \frac{3}{4} \cdot \frac{1}{3} = \frac{1}{4}\), you can confidently say that \(A\) and \(B\) are independent. Understanding this independence allows for easier calculations and predictions, as the relationship between events is clearly defined.

Probability formulas

There are key formulas in probability that help solve problems involving events and their likelihoods. These formulas allow us to compute probabilities systematically and with precision.

• Complement Formula: \(P(\bar{A}) = 1 - P(A)\) helps us find the chance that an event does not happen.
• Addition Formula: The formula \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\) calculates the probability that either of two events (or both) happen.
• Multiplication for Independence: \(P(A \cap B) = P(A) \cdot P(B)\) when \(A\) and \(B\) are independent shows joint probability.

These formulas are the building blocks of probability calculations. They are vital for determining specific outcomes and understanding how various events interact or coexist. When using these formulas, careful substitution of known values leads to a clear understanding of probability scenarios, as seen in solving the given exercise. Employing these tools effectively can simplify even complex probability problems.