Problem 14 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(\mathrm{A}\) and \(\mathrm{B}\) be two non-null events such that \(\mathrm{A} \subset \mathrm{B}\). Then, which of the following statements is always correct? [April \(08,2019(\) I) \(]\) (a) \(\mathrm{P}(\mathrm{A} \mid \mathrm{B})=\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A})\) (b) \(\mathrm{P}(\mathrm{A} \mid \mathrm{B}) \geq \mathrm{P}(\mathrm{A})\) (c) \(\mathrm{P}(\mathrm{A} \mid \mathrm{B}) \leq \mathrm{P}(\mathrm{A})\) (d) \(\mathrm{P}(\mathrm{A} \mid \mathrm{B})=1\)

Step-by-Step Solution

Answer

The correct statement is (d) \(P(A \mid B) = 1\).

Step 1: Understanding the Problem

We are given two events, \(A\) and \(B\), such that \(A\) is a subset of \(B\) (\(A \subset B\)). We need to determine which statement about conditional probabilities is always true based on this relationship.

Step 2: Reviewing the Concept of Conditional Probability

The conditional probability \(P(A \mid B)\) represents the probability of event \(A\) occurring given that event \(B\) has occurred. It is calculated using \(P(A \mid B) = \frac{P(A \cap B)}{P(B)}\).

Step 3: Applying the Subset Property

Given \(A \subset B\), it follows that \(A \cap B = A\), because everything in \(A\) is also in \(B\). Thus, \(P(A \cap B) = P(A)\). Therefore, the conditional probability becomes \(P(A \mid B) = \frac{P(A)}{P(B)}\).

Step 4: Evaluating Each Statement

We evaluate each option:- (a) \(P(A \mid B) = P(B) - P(A)\): This is incorrect because subtraction doesn't represent conditional probability.- (b) \(P(A \mid B) \geq P(A)\): Substitute \(P(A \mid B) = \frac{P(A)}{P(B)}\), this is not universally true.- (c) \(P(A \mid B) \leq P(A)\): Substitute \(P(A \mid B) = \frac{P(A)}{P(B)}\), this isn't always true either.- (d) \(P(A \mid B) = 1\): Since \(A\) is a subset of \(B\), when \(B\) occurs, \(A\) always occurs, making this statement true: \(P(A \mid B) = 1\).

Step 5: Conclusion

Based on the properties of subsets and conditional probabilities, the only correct statement is (d) \(P(A \mid B) = 1\). This is because whenever \(B\) occurs, \(A\) must also occur, given that \(A \subset B\).

Key Concepts

Subset

In probability, a subset is a smaller set of elements that are all contained within a bigger set. Let's say you have a set \(B\) that represents outcomes of a specific event. If set \(A\) is a subset of \(B\) (denoted as \(A \subset B\)), it means every outcome in \(A\) is also in \(B\). However, not every outcome in \(B\) must be in \(A\). This concept is crucial because it helps us understand relationships between events, especially when calculating probabilities. For conditional probability, the concept of subsets allows us to determine the outcomes involved more clearly. In terms of probability, when event \(A\) is a subset of event \(B\), we can say for certain when \(B\) happens, \(A\) happens as well because all elements of \(A\) are inside \(B\). This forms the basis for many probability discussions, such as those involving dependent events or conditional probabilities.

Probability

Probability is the measure of how likely an event is to occur. It ranges from 0, which means an event is impossible, to 1, meaning it is certain. To express probability as a mathematical notation, we use \(P(A)\) to denote the probability of event \(A\) occurring. Now when considering events like \(A\) and \(B\), where one is a subset of the other, probability helps determine expectations about the outcome. Probability can be calculated using a ratio of favorable outcomes to the total number of outcomes. For example, if you consider rolling a six-sided die, the probability of rolling a 3 is \(\frac{1}{6}\), because there is one favorable outcome and six possible outcomes. Understanding the foundational principles of probability makes further concepts like conditional probability easier to grasp as it builds upon these basic ideas.

Conditional Probability Formula

Conditional probability is a concept that measures how likely an event is, given that another event has already taken place. This is where the conditional probability formula comes in handy. It is defined as:

• \(P(A \mid B) = \frac{P(A \cap B)}{P(B)}\)

This formula computes the probability of event \(A\) given that event \(B\) occurs. The numerator \(P(A \cap B)\) represents the probability that both \(A\) and \(B\) occur, while the denominator \(P(B)\) represents the probability that \(B\) occurs. When \(A\) is a subset of \(B\), \(A \cap B\) simplifies to \(A\), because all elements of \(A\) are contained in \(B\). This makes it easier to determine the conditional probability that \(A\) occurs given \(B\) has already occurred.

Events in Probability

In probability theory, events are the outcomes or results that we are concerned about. Each event has a probability associated with it, which tells us how likely this event will occur. Events can interact in various ways such as independently, mutually exclusive, or through a conditional relationship. Understanding these interactions helps in calculating probabilities accurately.

• Independent Events: Two events are independent if the occurrence of one does not affect the probability of the other. For example, flipping two separate coins.
• Mutually Exclusive Events: These events cannot occur at the same time. For instance, rolling a 3 and a 4 on a single die throw.
• Conditional Events: Conditional probability deals with events that do affect each other, or are dependent. Such is the case when one event is a subset of another, which significantly simplifies calculations, such as that described in our initial example where \(A \subset B\).

Grasping how events relate to each other aids in mastering probability concepts, ultimately making it easier to solve complex probability tasks.