Problem 125 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(A\) and \(B\) are any two events, the probability that exactly one of them occurs is (A) \(P(A)+P(B)-2 P(A \cap B)\) (B) \(P(\bar{A})+P(\bar{B})-2 P(\bar{A} \cap \bar{B})\) (C) \(P(A \cap(\bar{B}))+P((\bar{A}) \cap B)\) (D) \(P(A)+P(B)-P(A \cup B)\)

Step-by-Step Solution

Answer

Option C: \(P(A \cap \bar{B}) + P(\bar{A} \cap B)\).

Step 1: Understand the Problem

We are asked to find the probability that exactly one of the events \(A\) or \(B\) occurs. This means we need to find the probability of either \(A\) occurring and \(B\) not occurring or \(B\) occurring and \(A\) not occurring.

Step 2: Set Up the Formula for Exactly One Event Occurring

The probability that exactly one of \(A\) or \(B\) occurs can be found by adding the probability of \(A\) occurring while \(B\) does not, and the probability of \(B\) occurring while \(A\) does not. This is written as:\[ P(A \cap \bar{B}) + P(\bar{A} \cap B) \]

Step 3: Verify the Options

Option (C) gives us precisely the expression we derived: \[ P(A \cap \bar{B}) + P(\bar{A} \cap B) \] This expression is exactly what we have determined as the probability of exactly one event occurring.

Key Concepts

Events in Probability

Understanding events in probability is fundamental to solving problems associated with the likelihood of occurrences. An event, in probability terms, refers to a set of outcomes resulting from a specific experiment. For example, when a coin is flipped, two possible events can occur: landing on heads or tails. Each event represents one possible outcome, and together they encompass the sample space of the experiment.

Events can be of different types, such as independent events, where the occurrence of one event does not affect the other, or mutually exclusive events, where the occurrence of one event precludes the occurrence of the other. Understanding these distinctions helps in assessing how events interact, which is crucial in probability calculations.

Set Operations in Probability

Set operations are key tools in probability that help us understand relationships between different events. The main operations include union, intersection, and complement.

• Union (P(A \cup B)): This operation represents the probability that at least one of the two events, \(A\) or \(B\), occurs. It's like combining the sets of possible outcomes from both events.
• Intersection (P(A \cap B)): This indicates the set of outcomes that both \(A\) and \(B\) share. If both events have common outcomes, their intersection shows this overlap.
• Complement (P(\bar{A})): The complement of an event \(A\) includes all outcomes in the sample space that are not \(A\). It helps in identifying probabilities of non-occurrence of an event.

These operations lay the foundation for calculating various probabilities, like the probability of exactly one event occurring, which involves identifying the outcomes exclusively associated with each event.

Probability of Events Occurring Exactly Once

When determining the probability of one out of two events occurring exactly once, we focus on distinct scenarios where one event happens and the other does not. This specific calculation avoids scenarios where both events occur simultaneously.

To find this, consider two events, \(A\) and \(B\). The expression for the probability of exactly one event occurring combines two probabilities: \(A\) occurring without \(B\), and \(B\) occurring without \(A\). Mathematically, this is expressed as:\[ P(A \cap \bar{B}) + P(\bar{A} \cap B) \]

This formula captures the essence of exclusivity — ensuring that only one of the two possible events happens. By precisely understanding the set operations involved, we isolate the needed outcomes, which is crucial in accurately determining such probabilities.