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