Problem 123 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(A\) and \(B\) are two events such that \(P(A \cup B) \geq \frac{3}{4}\) and \(\frac{1}{8} \leq P(A \cap B) \leq \frac{3}{8}\), then (A) \(P(A)+P(B) \leq \frac{11}{8}\) (B) \(P(A) \cdot P(B) \leq \frac{3}{8}\) (C) \(P(A)+P(B) \geq \underline{7}\) (D) none of these

Step-by-Step Solution

Answer

The correct option is (D) none of these.

Step 1: Understanding Given Information

We are given two events, \(A\) and \(B\), with the probabilities \(P(A \cup B) \geq \frac{3}{4}\) and \(\frac{1}{8} \leq P(A \cap B) \leq \frac{3}{8}\). We need to find which statement about \(P(A)\) and \(P(B)\) is true.

Step 2: Using the Addition Rule for Probabilities

The addition rule for probabilities gives us \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). We know \(P(A \cup B) \geq \frac{3}{4}\). Therefore, \(P(A) + P(B) - P(A \cap B) \geq \frac{3}{4}\).

Step 3: Substituting the Range of \(P(A \cap B)\)

Substitute the boundary condition shown for \(P(A \cap B)\) into the inequality:1. \(P(A) + P(B) - \frac{3}{8} \geq \frac{3}{4}\)2. \(P(A) + P(B) - \frac{1}{8} \geq \frac{3}{4}\)Simplifying these inequalities will give us the range for \(P(A) + P(B)\).

Step 4: Simplifying the First Inequality

From the inequality \(P(A) + P(B) - \frac{3}{8} \geq \frac{3}{4}\), we simplify it:\(P(A) + P(B) \geq \frac{3}{4} + \frac{3}{8} \)Converting \(\frac{3}{4}\) to \(\frac{6}{8}\), we get:\(P(A) + P(B) \geq \frac{9}{8}\).

Step 5: Simplifying the Second Inequality

From the inequality \(P(A) + P(B) - \frac{1}{8} \geq \frac{3}{4}\), we simplify it:\(P(A) + P(B) \geq \frac{3}{4} + \frac{1}{8}\)Converting \(\frac{3}{4}\) to \(\frac{6}{8}\), we get:\(P(A) + P(B) \geq \frac{7}{8}\).

Step 6: Identifying the Correct Statement

Comparing \(P(A) + P(B)\) from both inequalities, we have:\(\frac{9}{8} \geq P(A) + P(B) \geq \frac{7}{8}\).Since neither statement precisely fits this result, correct choice should be (D) none of these based on options available.

Key Concepts

Addition Rule of Probability

When solving probability problems, one often encounters multiple events happening together. Here, the Addition Rule of Probability comes into play. This fundamental rule helps us find the probability of either one event or another, or both, occurring again. If you are wondering what these events are, consider the union of two events. The mathematical expression for this is:

• \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)

Now, let’s break it down:
- \( P(A \cup B) \) represents the probability of event \(A\) or event \(B\) occurring.
- \( P(A) \) and \( P(B) \) are the individual probabilities of events \(A\) and \(B\) happening separately.
- \( P(A \cap B) \) accounts for the overlap – situations where both \(A\) and \(B\) occur simultaneously.
By substituting the given probabilities into this formula, we can easily calculate the desired outcomes in our problems, such as the one given above.

Events in Probability

In probability theory, an event refers to any possible outcome or a specific set of outcomes of a random process. For instance, when tossing a coin, the possible events can be getting heads or tails. Understanding how events are defined is crucial as it forms the base of all probability calculations.
Experts categorize events into:

• Simple Events: Events with a single outcome, like rolling a three on a die.
• Compound Events: These include multiple simple events, like rolling an even number on a die (2, 4, or 6).

In problems involving multiple events, analyzing given probabilities and their interrelations is essential. Using equations and conditions provided, one can determine the actual likelihood of certain events occurring together or separately. Getting these foundations right is key when engaging with any probability problem.

Inequalities in Probability

Inequalities arise in probability when we seek to determine ranges or bounds of the probability of events. They are particularly useful when dealing with questions involving uncertain outcomes and limited information.
In our context, we've seen how inequalities help establish boundaries for complex expressions like \( P(A \cup B) \) or \( P(A \cap B) \). Consider the given conditions:

• \( P(A \cup B) \geq \frac{3}{4} \)
• \( \frac{1}{8} \leq P(A \cap B) \leq \frac{3}{8} \)

These inequalities help us derive meaningful bounds for \( P(A) + P(B) \), simplifying complex problems into manageable steps. They give insight, allowing us to exclude unreasonable probability values and guiding us toward more accurate results when evaluating conditions and solving probability equations.