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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 $\mathrm{E}$ and $\mathrm{F}$ independent ?
$E:$ ' the card drawn is a king and queen '
$F:$ ' the card drawn is a queen or jack '
Solution
In a deck of $52$ cards, $4$ cards are kings, $4$ cards are queens, and $4$ cards are jacks.
$\therefore \mathrm{P}(\mathrm{E})=\mathrm{P}$ (The card drawn is a king or a queen) $=\frac{8}{52}=\frac{2}{13}$
$\therefore \mathrm{P}(\mathrm{F})=\mathrm{P}$ (The card drawn is a king or a jack) $ =\frac{8}{52}=\frac{2}{13}$
There are $4$ cards which are king and queen or jack.
$\therefore $ $\mathrm{P}(\mathrm{EF})=\mathrm{P}$ (The card drawn is king or a queen, or queen or a jack) $=\frac{4}{52}=\frac{1}{13}$.
$\mathrm{P}(\mathrm{E}) \times \mathrm{P}(\mathrm{F})=\frac{2}{13} \cdot \frac{2}{13}=\frac{4}{169} \neq \frac{1}{13}$
$\Rightarrow \mathrm{P}(\mathrm{E}), \mathrm{P}(\mathrm{F}) \neq \mathrm{P}(\mathrm{EF})$
Therefore, the given events $E$ and $F$ are not independent.
