Events A, B, C are mutually exclusive events such that and and The set of possible values of x is in the interval

Engineering

Mathematics

Addition Theorem on Probability

Basic Definitions Related to Probability

Question

Events A, B, C are mutually exclusive events such that $P (A) = \frac{3 x + 1}{3}$ and $P (B) = \frac{1 - x}{4}$ and $P (C) = \frac{1 - 2 x}{2} .$ The set of possible values of x is in the interval

$[\frac{1}{3}, \frac{1}{2}]$

[0, 1]

$[\frac{1}{3}, \frac{13}{3}]$

$[\frac{1}{3}, \frac{2}{3}]$

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$P (A) = \frac{3 x + 1}{3}, P (B) = \frac{1 - x}{4}, P (C) = \frac{1 - 2 x}{2}$

$∴ 0 \leq P \leq 1$

$0 \leq \frac{3 x + 1}{3} \leq 1$

$\Rightarrow 0 \leq 3 x + 1 \leq 3$

$\frac{- 1}{3} \leq x \leq \frac{2}{3}$

similarly $0 \leq \frac{1 - x}{4} \leq 1$

$\Rightarrow - 4 \leq x - 1 \leq 0$

$\Rightarrow - 3 \leq x \leq 1$

$similarly$

$0 \leq \frac{1 - 2 x}{2} \leq 1 \Rightarrow - 2 \leq 2 x - 1 \leq 0$

$\frac{- 1}{2} \leq x \leq \frac{1}{2}$

$Also 0 \leq P(AUBUC) \leq 1$

$\Rightarrow 0 \leq P(A) + P(B) + P(C) - P(A⋂B) - P(B⋂C) - P(C⋂A) + P(A⋂B⋂C) \leq 1$

$\Rightarrow 0 \leq P(A) + P(B) + P(C) \leq 1$
Since A , B and C are mutually exhaustive hence P(A⋂B) = P(B⋂C) = P(C⋂A) P(A⋂B⋂C) = 0
$0 \leq \frac{3 x + 1}{3} + \frac{1 - x}{4} + \frac{1 - 2 x}{2} \leq 1$

$0 \leq \frac{13 - 3 x}{12} \leq 1$

$\Rightarrow - 12 \leq 3 x - 13 \leq 0$

$\frac{1}{3} \leq x \leq \frac{13}{3}$
comparing all the inequalities of x and taking the intersection we get

$\frac{1}{3} \leq x \leq \frac{1}{2} x \in [\frac{1}{3}, \frac{1}{2}]$