Problem 2 - 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) \(P(A)+P(B) \geq \frac{7}{8}\) (D) none of these

Step-by-Step Solution

Answer

The correct answer is (A) \(P(A)+P(B) \leq \frac{11}{8}\).

Step 1: Identify Given Conditions

We are given that \(P(A \cup B) \geq \frac{3}{4}\) and \(\frac{1}{8} \leq P(A \cap B) \leq \frac{3}{8}\). These inequalities will help us determine the relationship between \(P(A)\) and \(P(B)\).

Step 2: Apply Formula for Union of Events

The formula for the probability of the union of two events is \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). Rearranging it gives us \(P(A) + P(B) = P(A \cup B) + P(A \cap B)\).

Step 3: Substitute Known Inequalities

Substitute the known inequalities into the rearranged formula: \(P(A) + P(B) = P(A \cup B) + P(A \cap B)\). Using the least values, \(P(A) + P(B) \geq \frac{3}{4} + \frac{1}{8} = \frac{7}{8}\).

Step 4: Evaluate the Other Inequality

Using the greatest values gives \(P(A) + P(B) \leq P(A \cup B) + P(A \cap B) \leq 1 + \frac{3}{8} = \frac{11}{8}\).

Step 5: Assess Options Against Results

From our analysis, both (A) and (C) appear to be true based on different substitutions. However, option (A) is directly confirmed by \(P(A) + P(B) \leq \frac{11}{8}\).

Step 6: Conclusion

Comparing our results to the options shows that option (A) \(P(A)+P(B) \leq \frac{11}{8}\) is consistent with the calculated inequalities.

Key Concepts

Union of Events

In probability theory, the concept of the "Union of Events" refers to the event that at least one of several events occurs. When we talk about the union of two events, say \(A\) and \(B\), we are considering any scenario where either event \(A\) happens, event \(B\) happens, or both events happen simultaneously.

Mathematically, the probability of the union is denoted as \(P(A \cup B)\). The formula to calculate this probability is given by:

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

This formula accounts for the basic addition of probabilities but subtracts the intersection part \(P(A \cap B)\) because it gets counted twice when we simply add \(P(A)\) and \(P(B)\).

In our exercise, we know that \(P(A \cup B) \geq \frac{3}{4}\), which is a significant piece of information. This tells us that there's a high probability that either event \(A\), event \(B\), or both happen.

Intersection of Events

"Intersection of Events" focuses on the scenario where two events happen simultaneously. In probability terms, if we consider events \(A\) and \(B\), the intersection, denoted as \(P(A \cap B)\), measures the probability that both \(A\) and \(B\) occur together.

The range for \(P(A \cap B)\) in our exercise is given as \(\frac{1}{8} \leq P(A \cap B) \leq \frac{3}{8}\). This range is essential because it provides limits within which the probability of both events happening lies.

Understanding this concept ensures that when analyzing the overall probabilities involving \(A\) and \(B\), we can accurately adjust for when they're mutually dependent or overlap. In simpler terms, when both events occur, this value reduces the over-counting in the union formula.

Probability Inequality

Probability inequalities provide us with bounds and relationships between different probabilities. They help in estimating or comparing probabilities when exact figures are not possible to determine.

In the given exercise, we see examples of probability inequalities. For instance:

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

By rearranging and substituting these inequalities in the union and intersection formulas, we derive larger or smaller bounds for different probabilities, such as \(P(A) + P(B)\).

Using these inequalities helps us understand the probability landscape between events like \(A\) and \(B\) without knowing each probability in isolation. They provide insights into how probabilities are distributed across individual and joint events, guiding us to determine the plausibility of various scenarios or outcomes.