Question:
Let $E$ and $F$ be events with $P(E)=\frac{3}{5}, P(F)=\frac{3}{10}$ and $P(E \cap F)=\frac{1}{5} .$ Are $E$ and $F$ independent?
Solution:
It is given that $\mathrm{P}(\mathrm{E})=\frac{3}{5}, \mathrm{P}(\mathrm{F})=\frac{3}{10}$, and $\mathrm{P}(\mathrm{EF})=\mathrm{P}(\mathrm{E} \cap \mathrm{F})=\frac{1}{5}$
$\mathrm{P}(\mathrm{E}) \cdot \mathrm{P}(\mathrm{F})=\frac{3}{5} \cdot \frac{3}{10}=\frac{9}{50} \neq \frac{1}{5}$
$\Rightarrow \mathrm{P}(\mathrm{E}) \cdot \mathrm{P}(\mathrm{F}) \neq \mathrm{P}(\mathrm{EF})$
Therefore, E and F are not independent.
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