One card is drawn at random from a well shuffled deck of $52$ cards. In which of the following cases are the events $E$ and $F$ independent ?

$\mathrm{E}:$  ' the card drawn is black ' 

$\mathrm{F}:$  ' the card drawn is a king '

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In a deck of $52$ cards, $26$ cards are black and $4$ cards are kings.

$\therefore $ $\mathrm{P}(\mathrm{E})=\mathrm{P}$ (the card drawn is a black ) $=\frac{26}{52}=\frac{1}{2}$

$\therefore $ $\mathrm{P}(\mathrm{F})=\mathrm{P}$ (the card drawn is a king ) $=\frac{4}{52}=\frac{1}{13}$

In the pack of $52$ cards, $2$ cards are black as well as kings.

$\therefore $ $\mathrm{P}(\mathrm{EF})=\mathrm{P}$ (the card drawn is  black king ) $=\frac{2}{52}=\frac{1}{26}$

$\mathrm{P}(\mathrm{E}) \times \mathrm{P}(\mathrm{F})=\frac{1}{2} \cdot \frac{1}{13}=\frac{1}{26}=\mathrm{P}(\mathrm{EF})$

Therefore, the given events $\mathrm{E}$ and $\mathrm{F}$ are independent.

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