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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 '
Solution
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.
