Problem 158 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(A\) and \(B\) be two events such that \(P(\overline{A \cup B})=\frac{1}{6}, P(A \cap B)=\frac{1}{4}\) and \(P(\bar{A})=\frac{1}{4}\), where \(\bar{A}\) stands for complement of event \(A\). Then events \(A\) and \(B\) are (A) equally likely and mutually exclusive (B) equally likely but not independent (C) independent but not equally likely (D) mutually exclusive and independent

Step-by-Step Solution

Answer

The correct option is (C) independent but not equally likely.

Step 1: Understand the Problem

We need to determine the relationship between two events, \(A\) and \(B\), given certain probabilities. The task requires identifying whether the events are equally likely, mutually exclusive, or independent.

Step 2: Use the Probability Formula

The given information includes: \(P(\overline{A \cup B}) = \frac{1}{6}\), \(P(A \cap B) = \frac{1}{4}\), and \(P(\bar{A}) = \frac{1}{4}\). First, let's find \(P(A \cup B)\), where \(\overline{A \cup B}\) is the complement:\[P(A \cup B) = 1 - P(\overline{A \cup B}) = 1 - \frac{1}{6} = \frac{5}{6}.\]

Step 3: Relationship Between \(A\) and \(B\)

We've identified \(P(A \cap B) = \frac{1}{4}\). Confirm their potential dependency using the formula:\[P(A \cup B) = P(A) + P(B) - P(A \cap B).\]Substituting into the equation:\[\frac{5}{6} = P(A) + P(B) - \frac{1}{4}.\]

Step 4: Find \(P(A)\) from Complement

Given \(P(\bar{A}) = \frac{1}{4}\), then the probability of event \(A\) occurring is\[P(A) = 1 - P(\bar{A}) = 1 - \frac{1}{4} = \frac{3}{4}.\]

Step 5: Substituting Values

Substitute \(P(A)\) into the equation to solve for \(P(B)\):\[\frac{5}{6} = \frac{3}{4} + P(B) - \frac{1}{4}.\]This simplifies to:\[\frac{5}{6} = \frac{1}{2} + P(B).\]Thus,\[P(B) = \frac{5}{6} - \frac{1}{2} = \frac{1}{3}.\]

Step 6: Check for Independence

For events \(A\) and \(B\) to be independent, \(P(A \cap B) = P(A) \cdot P(B)\).Checking:\[P(A) \cdot P(B) = \frac{3}{4} \cdot \frac{1}{3} = \frac{1}{4}.\]Since this equals \(P(A \cap B)\), the events are independent.

Step 7: Determine Equality of Likelihood

Two events are equally likely if \(P(A) = P(B)\). Since \(P(A) = \frac{3}{4}\) and \(P(B) = \frac{1}{3}\), they are not equally likely.

Step 8: Check for Mutual Exclusivity

Events are mutually exclusive if \(P(A \cap B) = 0\). Given \(P(A \cap B) = \frac{1}{4}\), the events are not mutually exclusive.

Key Concepts

Independent Events

In probability, two events are independent if the occurrence of one does not affect the occurrence of the other. This means that the probability of both events happening together, denoted by \(P(A \cap B)\), is equal to the product of their individual probabilities. Mathematically, this is represented as: \[ P(A \cap B) = P(A) \cdot P(B) \]In the given exercise, we determined that \(P(A \cap B) = \frac{1}{4}\). To check if events \(A\) and \(B\) are independent, we found:\[P(A) \cdot P(B) = \frac{3}{4} \cdot \frac{1}{3} = \frac{1}{4}\]Since this is equal to the given \(P(A \cap B)\), events \(A\) and \(B\) are indeed independent. Therefore, whether one event occurs does not impact the likelihood of the other happening. This is an essential concept because understanding independence helps in predicting the combined probability in future scenarios reliably.

Mutually Exclusive Events

Mutually exclusive events are those that cannot happen at the same time. If one of the events occurs, the other cannot. In probability terms, the intersection of two mutually exclusive events is zero: \[ P(A \cap B) = 0 \]In the exercise problem, we found that \(P(A \cap B) = \frac{1}{4}\). Clearly, this is not equal to zero, indicating that events \(A\) and \(B\) can occur simultaneously. Thus, they are not mutually exclusive. In real-life applications, understanding whether events are mutually exclusive is vital to avoid assuming that both can occur together, which helps in accurate probability calculations.

Equally Likely Events

Two events are equally likely if each has the same probability of occurring. This can be mathematically expressed by the equality:\[ P(A) = P(B) \]In our case, \(P(A) = \frac{3}{4}\) and \(P(B) = \frac{1}{3}\). Since these probabilities are not equal, events \(A\) and \(B\) are not equally likely. This is significant, as knowing the probability of events being equal can aid in making predictions where symmetry is important, like in games or designing fair processes.