Problem 26 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Events \(A, B, C\) are mutually exclusive events such that \(P(A)=\frac{3 x+1}{3}, P(B)=\frac{1-x}{4}\) and \(P(C)=\frac{1-2 x}{2}\) The set of possible values of \(x\) are in the interval. (a) \([0,1]\) (b) \(\left[\frac{1}{3}, \frac{1}{2}\right]\) (c) \(\left[\frac{1}{3}, \frac{2}{3}\right]\) (d) \(\left[\frac{1}{3}, \frac{13}{3}\right]\)

Step-by-Step Solution

Answer

Interval [1/3, 1/2].

Step 1: Understand Mutually Exclusive Events

Mutually exclusive events are events that cannot occur simultaneously. Therefore, the probability of the union of mutually exclusive events is the sum of their individual probabilities: \[ P(A \cup B \cup C) = P(A) + P(B) + P(C) \] For mutually exclusive events, it is important that their combined probability does not exceed 1, i.e., \[ P(A \cup B \cup C) \leq 1 \]

Step 2: Set Up the Inequality for Total Probability

Substitute the given probabilities into the equation: \[ P(A) + P(B) + P(C) = \frac{3x+1}{3} + \frac{1-x}{4} + \frac{1-2x}{2} \]This expression represents the total probability of the three mutually exclusive events.

Step 3: Simplify the Probability Expression

Combine the fractions: 1. Find a common denominator, which is 12.2. Convert each term: - \( \frac{3x+1}{3} = \frac{12x+4}{12} \) - \( \frac{1-x}{4} = \frac{3-3x}{12} \) - \( \frac{1-2x}{2} = \frac{6-12x}{12} \)3. Add these fractions:\[ \frac{12x+4}{12} + \frac{3-3x}{12} + \frac{6-12x}{12} = \frac{(12x+4) + (3-3x) + (6-12x)}{12} \]

Step 4: Further Simplification

Simplify the numerator:\[ 12x + 4 + 3 - 3x + 6 - 12x = 13 - 3x \]Therefore, the total probability simplifies to:\[ \frac{13 - 3x}{12} \]

Step 5: Set Up the Inequality

Since the probability must be at most 1, set up the inequality:\[ \frac{13 - 3x}{12} \leq 1 \]

Step 6: Solve the Inequality

Multiply through by 12 to clear the fraction:\[ 13 - 3x \leq 12 \]Rearrange to find 'x':\[ 1 \geq 3x \]\[ \frac{1}{3} \geq x \]

Step 7: Consider Non-negative Probabilities

Ensure that each probability is non-negative:1. \( \frac{3x+1}{3} \geq 0 \rightarrow x \geq -\frac{1}{3} \)2. \( \frac{1-x}{4} \geq 0 \rightarrow x \leq 1 \)3. \( \frac{1-2x}{2} \geq 0 \rightarrow x \leq \frac{1}{2} \)

Step 8: Combine Conditions

Combine all the conditions:\[ \frac{1}{3} \geq x \] and \[ x \leq 1 \] and \[ x \leq \frac{1}{2} \]The overlap of these intervals gives \( x \in \left[ \frac{1}{3}, \frac{1}{2} \right] \).

Key Concepts

Probability Inequality

Understanding probability inequality is essential when dealing with mutually exclusive events. Probability describes how likely an event is to occur, and inequality helps express constraints on possible values of these probabilities.
Within the context of mutually exclusive events like events \(A, B,\) and \(C\), there are specific rules to follow for the sum of their probabilities. Since they can't happen at the same time, the probability of any of them occurring is the sum of their individual probabilities: \[ P(A \cup B \cup C) = P(A) + P(B) + P(C) \]
However, the aggregate probability must never exceed 1. Consequently, the inequality \( P(A \cup B \cup C) \leq 1\) is applied to ensure that the total probability remains valid. This ensures all probabilities add up correctly in realistic terms.
Using inequalities, you can determine the range of values for a variable that affects probability values, which is crucial in probabilistic calculations like the one in the exercise.

Combined Probability

The concept of combined probability arises when multiple events are considered together. In the case of mutually exclusive events, the combined probability is the straightforward summation of individual event probabilities because no two events can happen in conjunction.
The standard equation for the combined probability becomes \( P(A \cup B \cup C) = P(A) + P(B) + P(C) \). By substituting the respective probabilities into this equation, you can simplify and solve for the combined likelihood of any one event happening.
For example, with probabilities given in fractional terms, simplifying the expressions may involve finding a common denominator, as shown in the step-by-step solution. Here, probabilities \(P(A)=\frac{3x+1}{3}\), \(P(B)=\frac{1-x}{4}\), and \(P(C)=\frac{1-2x}{2}\) were combined using a common denominator of 12 to find the overall probability expression \(\frac{13 - 3x}{12}\).
This combined probability should align with the principle that it must sum to no greater than 1, thus reflecting that the total probability stays within realistic limits.

Probability Intervals

Probability intervals are important in determining the range of values that a probability can take. For mutually exclusive events, they illustrate how all probabilities can coexist in a valid probability space, offering insight into variable constraints.
When we derive a constraint like \( \frac{1}{3} \geq x\), it tells us about a segment of possible values for \(x\), such that any sum of probabilities does not surpass 1. This is one vital aspect of understanding probability intervals.
Since individual probabilities must also be non-negative, additional inequalities are considered for each event’s probability:

• \(\frac{3x+1}{3} \geq 0\) implies \(x \geq -\frac{1}{3}\)
• \(\frac{1-x}{4} \geq 0\) implies \(x \leq 1\)
• \(\frac{1-2x}{2} \geq 0\) implies \(x \leq \frac{1}{2}\)

Combining these inequalities can pinpoint feasible values for variables in probability calculations. By considering them together, the interval \([\frac{1}{3}, \frac{1}{2}]\) is obtained, indicating that \(x\) must fall within this range to maintain valid and realistic probability values.