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 a spade'
$F:$ 'the card drawn is an ace'
$E:$ 'the card drawn is a spade'
$F:$ 'the card drawn is an ace'
(A) In a deck of $52$ cards,there are $13$ spades and $4$ aces.
$P(E) = P(\text{the card drawn is a spade}) = \frac{13}{52} = \frac{1}{4}$
$P(F) = P(\text{the card drawn is an ace}) = \frac{4}{52} = \frac{1}{13}$
There is only $1$ card that is both a spade and an ace (the ace of spades).
$P(E \cap F) = P(\text{the card drawn is a spade and an ace}) = \frac{1}{52}$
Since $P(E) \times P(F) = \frac{1}{4} \times \frac{1}{13} = \frac{1}{52} = P(E \cap F)$,the events $E$ and $F$ are independent.
$P(E) = P(\text{the card drawn is a spade}) = \frac{13}{52} = \frac{1}{4}$
$P(F) = P(\text{the card drawn is an ace}) = \frac{4}{52} = \frac{1}{13}$
There is only $1$ card that is both a spade and an ace (the ace of spades).
$P(E \cap F) = P(\text{the card drawn is a spade and an ace}) = \frac{1}{52}$
Since $P(E) \times P(F) = \frac{1}{4} \times \frac{1}{13} = \frac{1}{52} = P(E \cap F)$,the events $E$ and $F$ are independent.
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