Problem 163 - A Complete Resource Book in Mathematics for JEE Main

Question

A die is thrown. Let \(\mathrm{A}\) be the event that the number obtained is greater than \(3 .\) Let \(\mathrm{B}\) be the event that the number obtained is less than 5 . Then \(P(A \cup B)\) is \([2008]\) (A) \(\frac{3}{5}\) (B) 0 (C) 1 (D) \(\frac{2}{5}\)

Step-by-Step Solution

Answer

The probability \(P(A \cup B)\) is 1.

Step 1: Analyze Events

Identify the outcomes for each event. Event A: The outcomes where the number is greater than 3 are \(4, 5, 6\).Event B: The outcomes where the number is less than 5 are \(1, 2, 3, 4\).

Step 2: Define Union of Events

The union of events \(A \cup B\) includes all outcomes that are in either event A or event B or in both.From our analysis:\(A = \{4, 5, 6\}\), \(B = \{1, 2, 3, 4\}\).Thus, \(A \cup B = \{1, 2, 3, 4, 5, 6\}\).

Step 3: Calculate Probability for Union

Since a die has 6 sides, the total number of possible outcomes is 6.The set \(A \cup B\) includes all 6 outcomes. Therefore, the probability of the union is the number of favorable outcomes divided by the total number of outcomes: \(P(A \cup B) = \frac{6}{6} = 1\).

Key Concepts

Union of Events

When discussing probability, the term 'Union of Events' refers to the set of outcomes that belong to at least one of two events. Let's break it down with the given example of throwing a fair die. Consider two events: Event A (where the outcome is greater than 3) and Event B (where the outcome is less than 5). In probability, the symbol \(A \cup B\) denotes the union of events A and B.

• The union includes all outcomes that are members of A, or members of B, or members of both simultaneously.

In our example:

• Event A contains outcomes \( \{4, 5, 6\} \).
• Event B contains outcomes \( \{1, 2, 3, 4\} \).

To find the union, we combine these sets: \(A \cup B = \{1, 2, 3, 4, 5, 6\}\). This means any roll of the die falls under this union since it encapsulates all outcomes of a six-sided die. Understanding the union is critical as it often represents a common operation when evaluating chances for overlapping scenarios.

Sample Space

The sample space in probability is the set of all possible outcomes of a random experiment. It acts as the foundational backdrop upon which we examine specific events. In the case of a six-sided die, the sample space comprises all the numbers that can possibly appear on the top face of the die after it is rolled. Hence, the complete set, or the sample space, is:

• \(S = \{1, 2, 3, 4, 5, 6\}\)

Every time you roll the die, one of these numbers is the result. The concept of sample space is crucial because it establishes the universe of all outcomes and allows us to gauge the likelihood of various events. For example, if we want to assess an event described by a subset of the sample space, such as getting an even number, we know to focus only on \(\{2, 4, 6\}\). This structure supports many probability calculations, ensuring clarity and consistency in results.

Event Outcomes

'Event Outcomes' in probability refer to the specific results that satisfy the conditions of the event under consideration. When rolling a six-sided die, each number from 1 to 6 represents a potential outcome. When events are defined, such as Event A (obtaining a number greater than 3), we specify which of these outcomes satisfy the condition.In our example, we have:

• Event A: Outcomes \(\{4, 5, 6\}\)
• Event B: Outcomes \(\{1, 2, 3, 4\}\)

Understanding event outcomes helps us compute probabilities effectively. The probability of an event is the ratio of its favorable outcomes to the total number of outcomes in the sample space. This means to determine \(P(A)\) (the probability of Event A), we count its outcomes and divide by the total outcomes in the sample space: \(P(A) = \frac{3}{6} = \frac{1}{2}\). This same logic applies to all probability scenarios, underscoring the importance of distinguishing the outcomes of each event. Knowing these helps in understanding how events overlap, intersect, or unite, thereby facilitating accurate probability assessments.