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

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

None of the options are generally correct as presented unless additional conditions are assumed.

Step 1: Understanding Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. For two events, say \(C\) and \(D\), the conditional probability is given by the formula \[P(C \mid D) = \frac{P(C \cap D)}{P(D)}\]This tells us the probability of event \(C\) occurring when event \(D\) has already occurred.

Step 2: Using the Subset Property

Since \(C \subset D\), it means that whenever \(C\) occurs, \(D\) must also occur. Hence, \(C \cap D = C\). Therefore, the conditional probability can be rewritten as:\[P(C \mid D) = \frac{P(C)}{P(D)}\]

Step 3: Analyzing the Options

Let's analyze the given options:- **(A)** \(P(C \mid D) \geq P(C)\): We know \(P(C \mid D) = \frac{P(C)}{P(D)}\). Since \(P(D) \leq 1\), \(\frac{P(C)}{P(D)} \geq P(C)\) is unlikely unless \(P(D) = 1\).- **(B)** \(P(C \mid D) < P(C)\): Since \(P(D) \leq 1\), and if \(P(D) < 1\), then \(\frac{P(C)}{P(D)} > P(C)\).- **(C)** \(P(C \mid D) = \frac{P(D)}{P(C)}\): This does not follow from our derived equation.- **(D)** \(P(C \mid D) = P(C)\): This would imply \(P(D) = 1\), which may not be true in general.

Step 4: Selecting the Correct Option

From the analysis, **option (A)** is not generally valid as explained, and **option (B)** does not hold because \(P(C \mid D)\) is generally greater than or equal to \(P(C)\) given \(C \subset D\) and \(P(D)\) is not 1. **Option (D)** could potentially hold, but only if \(P(D) = 1\), which isn't the general case. Therefore, the potential correct statement is found when the inequality is reversed as shown in the initial argument.

Key Concepts

Subset Property

In probability theory, the subset property plays a significant role in understanding relationships between events. Let's consider two events, \(C\) and \(D\), where \(C\) is a subset of \(D\). This means that every time event \(C\) occurs, event \(D\) must also occur since all outcomes of \(C\) are contained within \(D\).
This is denoted as \(C \subset D\). The subset property helps in simplifying probability calculations as the intersection of \(C\) and \(D\), denoted as \(C \cap D\), is simply \(C\).
This simplification is crucial when calculating the conditional probability of \(C\) given \(D\). Understanding this property helps in making sound predictions and informed decisions based on probabilistic events.
Subset relationships are foundational in set theory and are pivotal in probability, as they allow us to determine conditional outcomes and relative likelihoods of intertwined events. The concept is visually similar to nesting dolls, where one effect completely resides within the other.

Event Probability

The probability of an event, often denoted as \(P(E)\), provides a measure of how likely that event is to occur. Probabilities range between 0 and 1, where 0 indicates impossibility, and 1 indicates certainty.
Consider an event \(C\), its probability \(P(C)\) tells you how confidently you can predict the occurrence of \(C\) out of all possible events in the sample space.
Likewise, for event \(D\), probability \(P(D)\) communicates the likelihood of \(D\) happening. Calculating event probabilities requires understanding the total number of potential outcomes versus the number of favorable outcomes.
While individual event probabilities are helpful, they become even more powerful when used in conjunction with conditional probability. When calculating probabilities for events that are subsets, like \(C \subset D\), you have better information to predict the likelihood of their co-occurrence. This interaction is central when working on combined events and in real-world applications where events are not independent.

Probability Theory

Probability theory is a branch of mathematics that deals with the analysis and interpretation of random events. It provides the framework for quantifying uncertainty and modeling the likelihood of various outcomes.
It is pivotal in fields such as statistics, finance, science, and engineering. The goal of probability theory is to assign numbers to the likelihood of events, enabling calculations and predictions to be systematically conducted.
Fundamental components of probability theory include:

• Probability spaces, which define the sample space, events, and probability measures.
• Random variables, functions that assign numerical values based on outcomes of random phenomena.
• Probability distributions, describing how probabilities are spread over values of a random variable.
• The laws of probability, such as addition and multiplication rules, conditional probability, and Bayes’ theorem.

These elements allow analysts and decision-makers to model uncertainty and derive insights from data. Conditional probability, a crucial aspect of probability theory, describes the probability of an event occurring provided that another event has already happened.
It helps in the understanding of dependencies between events and often relies on the subset property to simplify calculations. Mastery of probability theory facilitates the construction of models that depict real-world situations and contributes to informed decision-making.