Problem 18 - A Complete Resource Book in Mathematics for JEE Main

Question

If two events \(A\) and \(B\) are such that \(P\left(A^{2}\right)=0.3, P(B)\) \(=0.4\) and \(P\left(A B^{\circ}\right)=0.5\), then \(P\left[B \backslash\left(A \cup B^{c}\right)\right]=\) (A) \(\frac{1}{2}\) (B) \(\frac{1}{3}\) (C) \(\frac{1}{4}\) (D) none of these

Step-by-Step Solution

Answer

\( P(B \mid A \cup B') = \frac{1}{4} \). Answer: (C).

Step 1: Given data

\( P(A') = 0.3 \Rightarrow P(A) = 0.7 \), \( P(B) = 0.4 \), \( P(A \cap B') = 0.5 \). Find \( P(B \mid A \cup B') \).

Step 2: Find P(A ∩ B)

\( P(A \cap B) = P(A) - P(A \cap B') = 0.7 - 0.5 = 0.2 \).

Step 3: Find P(A ∪ B')

\( P(A \cup B') = P(A) + P(B') - P(A \cap B') = 0.7 + 0.6 - 0.5 = 0.8 \).

Step 4: Simplify B ∩ (A ∪ B')

\( B \cap (A \cup B') = (B \cap A) \cup (B \cap B') = A \cap B \). So \( P = 0.2 \).

Step 5: Answer

\( P(B \mid A \cup B') = 0.2/0.8 = 1/4 \). Answer: (C) \( \frac{1}{4} \).

Key Concepts

Set theory

Set theory is a mathematical framework for understanding collections of objects or elements typically called "sets." A set can be visualized like a container that holds multiple objects, things, or numbers. In probability theory, sets represent events or outcomes from an experiment.
For example, in our exercise, events denoted by sets like \(A\) and \(B\). Sets can undergo various operations, like intersections, unions, and complements. These operations help us understand complex probabilistic interactions. Understanding set theory is fundamental to analyzing probabilities, as it shows how different events relate to each other.

• An intersection (\(A \cap B\)) involves elements common to both sets \(A\) and \(B\).
• A union (\(A \cup B\)) contains all elements from either \(A\) or \(B\) or both.
• The complement (\(B^c\)) comprises elements not in \(B\).
• The set difference (\(B \backslash A\)) includes elements in \(B\) but not in \(A\).

By combining these basic operations, we explore complex events and calculate probabilities, enhancing our understanding of the scenario at hand.

Conditional probability

Conditional probability is a measure that calculates the probability of one event occurring, given that another event has already occurred. This concept captures how the occurrence of one event alters the likelihood of another.
In mathematical terms, the conditional probability of event \(A\) given event \(B\) is represented as \(P(A|B)\). It is calculated using the formula: \[P(A|B) = \frac{P(A \cap B)}{P(B)}\] This expresses the likelihood of \(A\) happening, assuming \(B\) happens as well.
Conditional probability helps understand dependent events, where one outcome affects the other.
For example, if you have information that someone is walking with an umbrella, the probability that it is raining might be higher. Understanding this concept is crucial for comprehensively solving problems in probability theory where event dependencies need to be considered.

Joint probability

Joint probability considers the probability of two or more events occurring simultaneously. It's vital in problems where the likelihood of multiple outcomes happening at the same time is analyzed.
For instance, joint probability calculates the chance of drawing a card from a deck that is both red and a queen, capturing the intersection of both conditions.
In mathematical notation, joint probability of events \(A\) and \(B\) is written as \(P(A \cap B)\).
This probability measures the concurrence of two occurring events.

• Independence: If \(A\) and \(B\) are independent, \(P(A \cap B) = P(A) \cdot P(B)\).
• Dependence: Joint probability requires knowing how the occurrence of one event affects another if they are not independent.

Being able to calculate joint probabilities allows one to predict and understand combined outcomes, significantly benefiting strategic decision-making in uncertainty.

Set operations

Set operations are crucial tools in the analysis of probability problems, as they allow manipulation and combination of different sets to examine complex relationships.
In probability theory, sets often represent different events, and operations on these sets help evaluate probabilities involving relationships like intersection, union, complement, and difference.
The most common operations include:

• Intersection (\(\cap\)) - Represents outcomes present in both sets.
  
• Union (\(\cup\)) - Includes all outcomes present in either of the sets.
  
• Complement (\(^c\)) - Contains outcomes not present in the set.
  
• Set difference (\(\backslash\)) - Consists of outcomes in one set but not in the other.

In the given problem, understanding set difference is crucial. It's used to figure out \(P[B \backslash (A \cup B^c)]\), defining events exclusive to \(B\) but absent in the union of \(A\) and the complement of \(B\).
Mastering set operations allows for nuanced solutions to probability problems, offering a clearer picture of how individual events relate within a broader system.