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Probability Basic — Applied Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceApplied MathsProbability Basic3 Marks
Question
If $E_1, E_2, E_3$ are three mutually exclusive events and exhaustive events of an experiment such that-
$2 P\left(E_1\right)=3 P\left(E_2\right)=P\left(E_3\right)$, then find $P\left(E_1\right)$.
✓

Answer

Since, $E_1, E_2, E_3$ are mutually exclusive and exhaustive events, so $E_1 \cap E_2=\phi, E_2 \cap E_3=\phi$, $E_1 \cap E_3=\phi, E_1 \cap E_2 \cap E_3=\phi$ and $E_1 \cup E_2 \cup E_3=S 1$ $\therefore \quad P\left(E_1 \cup E_2 \cup E_3\right)=P\left(E_1\right)+P\left(E_2\right)+P\left(E_3\right)$$-P\left(E_1 \cap E_2\right)-P\left(E_2 \cap E_3\right)$$-P\left(E_1 \cap E_3\right)+P\left(E_1 \cap E_2 \cap E_3\right)$
$\begin{array}{l}=P\left(E_1\right)+\frac{2}{3} P\left(E_1\right)+2 P\left(E_1\right) \\ =\frac{11}{3} P\left(E_1\right)\end{array}$
$\Rightarrow \quad P(S)=\frac{11}{3} P\left(E_1\right)$
$\Rightarrow \quad 1=\frac{11}{3} P\left(E_1\right)$
$\Rightarrow \quad P\left(E_1\right)=\frac{3}{11}$.

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