Problem 167 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(C\) and \(D\) are two events satisfying \(C \subset D\) with \(P(D)\) \(\neq 0\), then the correct statement among the following is [2011] (A) \(P(C \mid D) \geq P(C)\) (B) \(P(C \mid D)

Step-by-Step Solution

Answer

The correct statement is (A) \(P(C \mid D) \geq P(C)\).

Step 1: Understand Conditional Probability

Conditional probability is defined as the probability of an event occurring given that another event has already occurred. Mathematically, it is represented as \(P(A \mid B) = \frac{P(A \cap B)}{P(B)}\), where \(A\) and \(B\) are events and \(P(B) eq 0\).

Step 2: Analyze Given Information

Given that \(C \subset D\), it implies that every outcome in \(C\) is also in \(D\). Hence, \(C \cap D = C\) and therefore, \(P(C \cap D) = P(C)\).

Step 3: Calculate \(P(C \mid D)\)

Using the conditional probability formula \(P(C \mid D) = \frac{P(C \cap D)}{P(D)}\), and substituting from Step 2, we find \(P(C \mid D) = \frac{P(C)}{P(D)}\).

Step 4: Determine Correct Statement

Given that \(P(D) \geq P(C)\) because \(C \subset D\), we have \(\frac{P(C)}{P(D)} \leq P(C)\). Hence \(P(C \mid D) \leq P(C)\), making the correct statement an inequality scenario.

Key Concepts

Probability Theory

Probability Theory is the mathematical framework for quantifying the likelihood of different outcomes. Its core elements include events, outcomes, and probabilities.

• An event is a specific set of outcomes that we are interested in; for example, rolling a die and getting a number greater than four.
• An outcome is a single, distinct result, such as rolling a three on a die.
• The probability of an event is the measure of how likely it is to occur, usually expressed as a number between 0 and 1.

Breaking it down, Probability Theory helps us calculate the odds of certain events happening. By understanding how different outcomes can occur and linking them with their probabilities, one can make predictions or ascertain the likelihood of future events. In the context of the exercise, understanding probability theory, particularly conditional probability, allows us to relate different events to each other in terms of likelihood. This is crucial when events are not independent and depend on the occurrence of other events.

Events and Outcomes

In Probability Theory, each possible occurrence in a trial is known as an outcome, and a collection of outcomes is known as an event.

• For instance, flipping a coin could result in two possible outcomes: heads or tails.
• Suppose we are interested in a single event, such as rolling a number less than three with a six-sided die.

Connecting these concepts to the original exercise, we have two events, C and D. The fact that C is a subset of D means every outcome in C is also contained within D. This relationship implies that whenever event C occurs, event D must also occur. Hence, understanding these relationships allows us to draw conclusions about the probabilities associated with compound events and how they relate to each other.

Inequality in Probability

Inequalities in Probability often help us determine restrictive conditions or relationships between different probabilities. In the context of the exercise, we are dealing with an inequality:

• Given that C is a subset of D, it implies that event D is always more likely or equally likely to occur compared to event C.
• Since we have the relationship \(P(C \mid D) = \frac{P(C)}{P(D)}\), the constraint \(C \subset D\) ensures that \(P(D) \geq P(C)\).

Thus, the expression \(P(C \mid D) \leq P(C)\) indicates that the probability of event C occurring given D has already occurred will never exceed the standalone probability of event C itself. This concept highlights a foundational aspect of probability theory, where the likelihood of an event conditioned on another event's occurrence may be less than or equal to its actual likelihood, reinforcing the containment relationship between events C and D.