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?
$E:$ 'the card drawn is black'
$F:$ 'the card drawn is a king'
$E:$ 'the card drawn is black'
$F:$ 'the card drawn is a king'
(N/A) In a deck of $52$ cards,there are $26$ black cards and $4$ kings.
$P(E) = P(\text{the card drawn is black}) = \frac{26}{52} = \frac{1}{2}$
$P(F) = P(\text{the card drawn is a king}) = \frac{4}{52} = \frac{1}{13}$
In the pack of $52$ cards,there are $2$ cards that are both black and kings (the king of spades and the king of clubs).
$P(E \cap F) = P(\text{the card drawn is a black king}) = \frac{2}{52} = \frac{1}{26}$
Since $P(E) \times P(F) = \frac{1}{2} \times \frac{1}{13} = \frac{1}{26} = P(E \cap F)$,the events $E$ and $F$ are independent.
$P(E) = P(\text{the card drawn is black}) = \frac{26}{52} = \frac{1}{2}$
$P(F) = P(\text{the card drawn is a king}) = \frac{4}{52} = \frac{1}{13}$
In the pack of $52$ cards,there are $2$ cards that are both black and kings (the king of spades and the king of clubs).
$P(E \cap F) = P(\text{the card drawn is a black king}) = \frac{2}{52} = \frac{1}{26}$
Since $P(E) \times P(F) = \frac{1}{2} \times \frac{1}{13} = \frac{1}{26} = P(E \cap F)$,the events $E$ and $F$ are independent.
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