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Let $\mathbf{E}$ and $\mathbf{F}$ be the events with $\mathbf{P}(\mathbf{E})=\frac{3}{5}, P(F)=\frac{3}{10}$ and $P(E \cap F)=\frac{1}{5}$. Are $E$ and $F$ independent?
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Verified Answer
$P(E)=\frac{3}{5}, P(F)=\frac{3}{10}$,
$$
\therefore \quad \mathrm{P}(\mathrm{E}) \times \mathrm{P}(\mathrm{F})=\frac{3}{5} \times \frac{3}{10}=\frac{9}{50}
$$
$$
\mathrm{P}(\mathrm{E} \cap \mathrm{F}) \neq \mathrm{P}(\mathrm{E}) \times \mathrm{P}(\mathrm{F})
$$
$\therefore$ The event $\mathrm{A}$ and $\mathrm{B}$ are not independent.
$$
\therefore \quad \mathrm{P}(\mathrm{E}) \times \mathrm{P}(\mathrm{F})=\frac{3}{5} \times \frac{3}{10}=\frac{9}{50}
$$
$$
\mathrm{P}(\mathrm{E} \cap \mathrm{F}) \neq \mathrm{P}(\mathrm{E}) \times \mathrm{P}(\mathrm{F})
$$
$\therefore$ The event $\mathrm{A}$ and $\mathrm{B}$ are not independent.
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