Problem 71 - A Complete Resource Book in Mathematics for JEE Main

Question

The probabilites of three events \(A, B\) and \(C\) are \(P(A)=0.6, P(B)=0.4\) and \(P(C)=0.5 .\) If \(P(A \cup B)\) \(=0.8, P(A \cap C)=0.3, P(A \cap B \cap C)=0.2\) and \(P(A \cup\) \(B \cup C) \geq 0.85\), then (A) \(0.2 \leq P(B \cap C) \leq 0.35\) (B) \(0.5 \leq P(B \cap C) \leq 0.85\) (C) \(0.1 \leq P(B \cap C) \leq 0.35\) (D) none of these

Step-by-Step Solution

Answer

Option (A) is valid because it fits the derived bounds: \(0.2 \leq P(B \cap C) \leq 0.35\).

Step 1: Apply Inclusion-Exclusion Principle

To find the probability of the union of three events, use the inclusion-exclusion principle. The formula is \[ P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(A \cap C) + P(A \cap B \cap C) \].

Step 2: Substitute Known Values

Substitute the given probabilities into the inclusion-exclusion formula: \[ P(A \cup B \cup C) = 0.6 + 0.4 + 0.5 - P(A \cap B) - P(B \cap C) - 0.3 + 0.2 \].

Step 3: Simplify the Expression

Simplify the expression for \( P(A \cup B \cup C) \) as follows: \[ P(A \cup B \cup C) = 1.7 - P(A \cap B) - P(B \cap C) + 0.2 \].

Step 4: Use Given Constraint for Union

Given that \( P(A \cup B \cup C) \geq 0.85 \), set up the inequality: \[ 1.9 - P(A \cap B) - P(B \cap C) \geq 0.85 \].

Step 5: Solve for P(B ∩ C)

Isolate \( P(B \cap C) \) in the inequality: \[ P(B \cap C) \leq 1.05 - P(A \cap B) \]. Without the exact value of \( P(A \cap B) \), let's solve: \( 0 \leq P(B \cap C) \leq 1.05 \).

Step 6: Use Additional Constraints

From \( P(A \cap B \cap C) = 0.2 \), both \( P(A \cap C) \) and \( P(A \cap B) - P(A \cap B \cap C) \) bounds \( P(B \cap C) \) within constraints.

Step 7: Evaluate Answer Choices

Analyze each option and compare it with derived bounds: given maximum is \( 1.05 \).Option (A) within \([0.2, 0.35]\): validOption (B) over \([0.5, 0.85]\): not validOption (C) within \([0.1, 0.35]\): valid(D) none: incorrect

Key Concepts

Inclusion-Exclusion Principle

The Inclusion-Exclusion Principle is a fundamental concept in probability theory. It is used to calculate the probability of the union of multiple events. When working with three events named \(A\), \(B\), and \(C\), it helps us find how much overlap, or intersection, exists between these events.
By including individual probabilities and excluding overlapping probabilities, we avoid double-counting. The formula for three events is:

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

This formula may look complex at first, but breaking it down makes it easier. First, sum the individual probabilities of events \(A\), \(B\), and \(C\). This gives a rough idea of the total probability.
However, when events overlap, they add a portion of their probability that has already been counted by the individual event components. Thus, we subtract the probabilities of the intersections of each pair \((A \cap B, B \cap C, A \cap C)\), then add back the intersection of all three \(A \cap B \cap C\).
Using this approach ensures that we accurately account for every part of the events once and only once in calculating probabilities.

Probability of Events

In probability theory, understanding the likelihood of different events is key. An event can be defined as an outcome or a set of outcomes from a specific experiment or situation.
When we deal with probability, it is crucial to know these terms:

• Independent events: These are events where the occurrence of one does not affect the occurrence of the other. For example, flipping a coin and rolling a dice are two independent events.
• Dependent events: In contrast, these events are related so that the outcome or occurrence of one can affect the other. Selecting two cards from a deck without replacement would involve dependent events because the probability of the second draw changes after the first.
• Union of events \((A \cup B)\): Refers to the event that either \(A\) or \(B\) or both occur.
• Intersection of events \((A \cap B)\): Refers to the likelihood that both \(A\) and \(B\) occur simultaneously.

Each event has a probability value ranging from 0 (impossible event) to 1 (certain event). When solving problems, it is essential to understand these values and relationships to analyze and compare probabilities effectively.

Set Theory in Probability

Set theory is essential for understanding and solving problems in probability as it provides the framework for describing events. Events are considered sets in probability theory, and we use set operations to figure out complex relationships between them.
Three main set operations that play a big role in probability are:

• Union \((A \cup B)\) represents the elements that are in either set \(A\) or set \(B\) or in both.
• Intersection \((A \cap B)\) signifies the elements common to both sets \(A\) and \(B\).
• Complement (\(A^c\) or \(\overline{A}\)) consists of all elements not in set \(A\).

Visual aids like Venn diagrams can help when working with these concepts. A Venn diagram illustrates these sets and the operations between them, allowing us to "see" the probability relationships as overlapping circles.
Understanding these concepts helps in making calculations clearer, especially when applying the Inclusion-Exclusion Principle, to avoid over- or under-counting probabilities. Using set theory correctly allows for precise probability computation which is essential in analyzing how events relate to one another.