Problem 132 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(A\) and \(B\) are two events, then the probability that at most one of \(A, B\) occurs is (A) \(1-P(A \cap B)\) (B) \(P\left(A^{\prime}\right)+P\left(B^{\prime}\right)-P\left(A^{\prime} \cap B^{\prime}\right)\) (C) \(P\left(A^{\prime}\right)+P\left(B^{\prime}\right)+P(A \cup B)\) (D) none of these

Step-by-Step Solution

Answer

The probability that at most one of \(A\) or \(B\) occurs is (A) \(1 - P(A \cap B)\).

Step 1: Identify the Given Information

We are given two events, \(A\) and \(B\). Our goal is to find the probability that at most one of these events occurs.

Step 2: Express At Most One Occurrence

The event 'at most one of \(A\) or \(B\) occurs' includes the following scenarios: neither \(A\) nor \(B\) occurs, or exactly one of the events \(A\) or \(B\) occurs. Mathematically, this can be expressed as \(A^{\prime} \cap B^{\prime} \) or \((A \cap B^{\prime}) \cup (A^{\prime} \cap B)\).

Step 3: Calculate Probability Components

The probability of \(A^{\prime} \cap B^{\prime}\) is the chance that neither \(A\) nor \(B\) occurs. The probabilities for exactly one event occurring are \(P(A) - P(A \cap B)\) for event \(A\) only, and \(P(B) - P(A \cap B)\) for event \(B\) only.

Step 4: Use Probability Addition Rule

According to the addition rule for probabilities, for mutually exclusive events, we sum the probabilities. Thus, the probability of exactly one or no events occurring is: \[ P(A^{\prime} \cap B^{\prime}) + (P(A) - P(A \cap B)) + (P(B) - P(A \cap B)) \].

Step 5: Simplify the Expression

Using the formula for total probability, simplify: \(P(A^{\prime} \cap B^{\prime}) = 1 - P(A \cup B)\), then \((P(A) - P(A \cap B)) + (P(B) - P(A \cap B)) = P(A) + P(B) - 2P(A \cap B)\). Then the total simplifies further to: \[1 - P(A \cap B)\].

Step 6: Select the Correct Answer

From the choices, the expression \(1 - P(A \cap B)\) corresponds to option (A).

Key Concepts

Events and Outcomes

In probability theory, an **event** is a specific outcome or a set of outcomes of a random process. Each possible event in the sample space has an associated probability. These probabilities determine how likely each event is to occur.

• An **outcome** refers to a result of a single trial of a random process. For example, getting a 'heads' when you flip a coin is an outcome.
• Multiple outcomes together make an event. For instance, rolling an even number on a die (2, 4, or 6) is an event made of three outcomes.

Each time we consider probability, we focus on how events and outcomes interact. Understanding these interactions helps in predicting how often certain results can occur in probabilistic terms. For example, in the original exercise, events **A** and **B** represent different scenarios whose probabilities we are interested in.

Addition Rule for Probabilities

The **addition rule for probabilities** is a key concept in probability that helps us calculate the chances of either one event or another happening. This rule is particularly helpful when events have overlapping outcomes.The basic form of the rule when events are not mutually exclusive is given by:\[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]

• **\( P(A \cup B) \)**: Probability that either event **A** or event **B** or both occur.
• **\( P(A) \) & \( P(B) \)**: Individual probabilities of events **A** and **B** occurring.
• **\( P(A \cap B) \)**: Probability that both events occur simultaneously. We subtract this to adjust for the overlap, as it was counted twice.

In our exercise, understanding this rule enabled us to compute the combined probabilities of different scenarios where either **A**, **B**, or neither occurs.

Mutually Exclusive Events

**Mutually exclusive events** are outcomes that cannot occur simultaneously in a single trial. This concept is crucial in correctly calculating probabilities for events that do not have overlapping outcomes.Here are some characteristics:

• If events **A** and **B** are mutually exclusive, the probability of both occurring at the same time is 0: \( P(A \cap B) = 0 \).
• The addition rule simplifies for mutually exclusive events to \( P(A \cup B) = P(A) + P(B) \) since there is no overlap.

However, in the exercise we examined, **A** and **B** were not mutually exclusive, meaning there is a possibility they could occur together. That's why we needed the full version of the addition rule, including the overlapping probability adjustment. This distinction ensures we correctly compute the chance of one event occurring without inaccurately double-counting shared outcomes.