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

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

(C) \(P(A)+P(B) \geq \frac{7}{8}\) is correct.

Step 1: Understanding Probability of Union

We are given that \(P(A \cup B) \geq \frac{3}{4}\). Now recall, for any two events \(A\) and \(B\), the probability of their union is given by: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Our task is to analyze this expression in conjunction with the other given conditions.

Step 2: Analyzing Intersection Probability Boundaries

The problem states \(\frac{1}{8} \leq P(A \cap B) \leq \frac{3}{8}\). This range for \(P(A \cap B)\) allows us to see how it interacts with the required \(P(A) + P(B)\) values.

Step 3: Deriving Relationship for $P(A)+P(B)$

Substitute the expression for \(P(A \cup B)\) into the inequality: \[ P(A) + P(B) - P(A \cap B) \geq \frac{3}{4} \] Applying the lower bound for \(P(A \cap B)\): \[ P(A) + P(B) - \frac{3}{8} \geq \frac{3}{4} \] Solving for \(P(A) + P(B)\), we get: \[ P(A) + P(B) \geq \frac{3}{4} + \frac{3}{8} = \frac{9}{8} \] Applying the upper bound of \(P(A \cap B)\): \[ P(A) + P(B) - \frac{1}{8} \geq \frac{3}{4} \] Solving, this results in: \[ P(A) + P(B) \geq \frac{7}{8} \] This shows (C) holds true given the constraints. Thus \(P(A) + P(B) \geq \frac{7}{8}\).

Step 4: Evaluating Other Options

Option (A) tests if the maximum value of \(P(A)+P(B)\) is \(\frac{11}{8}\). If \(P(A \cap B) = \frac{1}{8}\), \(P(A) + P(B) - \frac{1}{8} \geq \frac{3}{4}\) implies \(P(A)+P(B) \leq \frac{7}{8} + \frac{1}{8} = \frac{1}{8}\), which contradicts the finding of \(P(A) + P(B) \geq \frac{9}{8}\). Option (B) checks the product \(P(A)\cdot P(B)\). This requires detailed knowledge of individual probabilities and doesn't fit the computed union and intersection results as directly as (C).

Step 5: Conclusion and Verification

From the conditions given and tested, Option (C) holds. The inequality formulations verify \(P(A)+P(B) \geq \frac{7}{8}\) across the condition spectrum. This does not contradict any conditions.

Key Concepts

Union of Events

In probability theory, the union of events refers to the occurrence of at least one of the events happening. For two events, \(A\) and \(B\), the probability of their union, denoted \(P(A \cup B)\), represents the likelihood that either event \(A\), event \(B\), or both occur. This concept is key in scenarios where you want to determine the probability of either of multiple outcomes.
The formula to calculate the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
This formula helps us to avoid double-counting the probability of the intersection, where both events occur simultaneously. Understanding this relation is crucial since it forms the basis for optimal decision-making.
In the given problem, it is specified that \(P(A \cup B) \geq \frac{3}{4}\). This indicates that there's a high likelihood — at least 75% — that either or both events \(A\) and \(B\) will happen together.

Intersection of Events

The intersection of events examines the probability of two or more events happening at the same time. When we refer to the intersection of events \(A\) and \(B\), denoted by \(P(A \cap B)\), we are specifically looking at the case where both \(A\) and \(B\) occur simultaneously.
Knowing the intersection helps in understanding how correlated two events are. If the probability of their intersection is high, these events frequently occur together.
In the exercise, it's mentioned that \(\frac{1}{8} \leq P(A \cap B) \leq \frac{3}{8}\). This tells us that the probability of both events happening together is relatively low, ranging between 12.5% to 37.5%.
This range provides us with a boundary to calculate other probabilities, particularly when combined with the union of the same events. It aids in framing a clearer picture of how these events interact and affect each other's likelihood.

Probability Inequalities

Probability inequalities involve understanding the conditions and boundaries within which the probabilities of events lie. They are powerful tools in statistical and probability analysis, as they give constraints on what the probabilities can be, based on given or known data.
In this context, the calculated inequality for \(P(A) + P(B)\) in our problem shows the essential role of such inequalities.
Considering the given inequality and probability values, we determined the sum of \(P(A)\) and \(P(B)\):

• By utilizing the formula for the union of events: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\), the conditions provided help us establish: \(P(A) + P(B) \geq \frac{7}{8}\).
• This inequality gives a minimum combined probability for both events, derived considering the lower bounds of their intersection.

Probability inequalities require careful assessment of all intersecting probabilities to draw accurate conclusions on a range of values, ensuring no logical contradiction arises in the assumptions or the outcomes.