Problem 23 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If the events \(\mathrm{A}\) and \(\mathrm{B}\) are mutually exclusive events such that \(\mathrm{P}(\mathrm{A})=\frac{3 x+1}{3}\) and \(\mathrm{P}(\mathrm{B})=\frac{1-x}{4}\), then the set of possible values of \(x\) lies in the interval : \mathrm{\\{} O n l i n e ~ A p r i l ~ 2 5 , ~ 2 0 1 3 ] ~ (a) \([0,1]\) (b) \(\left[\frac{1}{3}, \frac{2}{3}\right]\) (c) \(\left[-\frac{1}{3}, \frac{5}{9}\right]\) (d) \(\left[-\frac{7}{9}, \frac{4}{9}\right]\)

Step-by-Step Solution

Answer

The set of possible values for \(x\) is \([-\frac{1}{3}, \frac{5}{9}]\).

Step 1: Understanding Mutually Exclusive Events

Mutually exclusive events are events that cannot happen at the same time. For events A and B, this means the probability of both occurring together is zero, i.e., \( P(A \cap B) = 0 \). Thus, \( P(A) + P(B) = 1 \).

Step 2: Setting Up the Equation

Given \( P(A) = \frac{3x + 1}{3} \) and \( P(B) = \frac{1-x}{4} \), use the property of mutually exclusive events to set up the equation: \[ \frac{3x + 1}{3} + \frac{1-x}{4} = 1 \].

Step 3: Solving the Equation

First, find a common denominator and simplify the equation: \[ \frac{4(3x + 1) + 3(1-x)}{12} = 1 \]. Simplify further: \[ \frac{12x + 4 + 3 - 3x}{12} = 1 \] which simplifies to \( \frac{9x + 7}{12} = 1 \). Solve for \(x\): \[ 9x + 7 = 12 \] which gives \( 9x = 5 \) and finally \( x = \frac{5}{9} \).

Step 4: Finding the Interval

Using the solved value of \(x\), check the interval options given. Note that the condition for \( P(A) \) and \( P(B) \) to be valid probabilities is \( 0 \leq P(A),P(B) \leq 1 \). Solve the inequalities \( \frac{3x + 1}{3} \geq 0 \) and \( \frac{1-x}{4} \geq 0 \). These resolve to \( x \geq -\frac{1}{3} \) and \( x \leq 1 \). By checking available options, \([0,\frac{5}{9}]\) is valid, leading to the intersection of these intervals to be \([-\frac{1}{3},\frac{5}{9}]\).

Key Concepts

Probability

Understanding probability is crucial when dealing with events that might happen, such as rolling a die or drawing a card. In probability, every event is assigned a likelihood of occurring, and this likelihood is expressed as a number between 0 and 1. For example, a probability of 0 means the event will not happen, while a probability of 1 means it will certainly occur.

In this exercise, we're discussing two mutually exclusive events, A and B. Since they are mutually exclusive, they can't both happen at the same time. This gives us a foundational rule to rely on: the sum of their probabilities is always 1. It's like saying if event A happens, event B can't use that opportunity space, and vice-versa. When working with these events in our mathematical expressions, it's important to ensure that our calculated probabilities, like \( P(A) \) and \( P(B) \), stay within the logical bounds of 0 and 1. This ensures that our real-world applications, like predicting outcomes, will be accurate.

Inequalities

Inequalities come into play when we need to determine the plausible range for a variable, such as \( x \), in our probability equations. These are tools that allow us to describe conditions in which certain mathematical expressions will hold true, leading to a range of possible solutions rather than just one.

For this exercise, we used inequalities to find acceptable values for the probabilities. Keeping \( 0 \leq P(A), P(B) \leq 1 \) in mind, we formed inequalities that the variable \( x \) must satisfy for these probabilities to remain valid. This step is crucial because even if an equation offers a numerical solution for \( x \), we must ensure that this solution maintains the overall conditions required by the problem. In practical terms, these inequalities guide us in finding the valid range of \( x \) that prevents violations of probability rules.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying expressions to solve equations efficiently. In this problem, it's vital in allowing us to combine multiple expressions and quantify unknowns, such as \( x \), by performing systematic steps.

Initially, having probabilities expressed in fractions can seem complex. By identifying a common denominator, we can add them together more easily. This exercise involves simplifying the expression \( \frac{3x + 1}{3} + \frac{1-x}{4} = 1 \) by finding a common denominator (12) to combine these fractions.

Through further rearrangements, we eventually obtained a linear equation, which we solved to find the specific value of \( x \). Simplification and cancellations along each step are key to reaching the solution straightforwardly. This often involves shifting parts of the equation to isolate variables on one side, simplifying to reach an answer. By distilling complex problems into solvable components, algebraic manipulation becomes a critical tool in not only solving the problem at hand but also comprehending the underlying mechanics of mathematical relationships.