Let $\mathrm{E}_{1}, \mathrm{E}_{2}, \mathrm{E}_{3}$ be three mutually exclusive events such that $$\mathrm{P}\left(\mathrm{E}_{1}\right)=\frac{2+3 \mathrm{p}}{6}, \mathrm{P}\left(\mathrm{E}_{2}\right)=\frac{2-\mathrm{p}}{8}$$ and $\mathrm{P}\left(\mathrm{E}_{3}\right)=\frac{1-\mathrm{p}}{2}$. If the maximum and minimum values of $\mathrm{p}$ are $\mathrm{p}_{1}$ and $\mathrm{p}_{2}$, then $\left(\mathrm{p}_{1}+\mathrm{p}_{2}\right)$ is equal to :
Solution
<p>$$0 \le {{2 + 3P} \over 6} \le 1 \Rightarrow P \in \left[ { - {2 \over 3},{4 \over 3}} \right]$$</p>
<p>$0 \le {{2 - P} \over 8} \le 1 \Rightarrow P \in [ - 6,2]$</p>
<p>$0 \le {{1 - P} \over 2} \le 1 \Rightarrow P \in [ - 1,1]$</p>
<p>$0 < P({E_1}) + P({E_2}) + P({E_3}) \le 1$</p>
<p>$0 < {{13} \over {12}} - {P \over 8} \le 1$</p>
<p>$P \in \left[ {{2 \over 3},{{26} \over 3}} \right]$</p>
<p>Taking intersection of all</p>
<p>$P \in \left[ {{2 \over 3},1} \right)$</p>
<p>${P_1} + {P_2} = {5 \over 3}$</p>
About this question
Subject: Mathematics · Chapter: Probability · Topic: Classical and Axiomatic Probability
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