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 king or a queen'
$F:$ 'The card drawn is a queen or a jack'

(NONE) In a deck of $52$ cards,there are $4$ kings,$4$ queens,and $4$ jacks.
$P(E) = P(\text{king or queen}) = \frac{4+4}{52} = \frac{8}{52} = \frac{2}{13}$.
$P(F) = P(\text{queen or jack}) = \frac{4+4}{52} = \frac{8}{52} = \frac{2}{13}$.
The event $E \cap F$ represents the card being a queen (since queen is common to both sets).
$P(E \cap F) = P(\text{queen}) = \frac{4}{52} = \frac{1}{13}$.
Now,check for independence: $P(E) \times P(F) = \frac{2}{13} \times \frac{2}{13} = \frac{4}{169}$.
Since $P(E \cap F) = \frac{1}{13} = \frac{13}{169}$,and $\frac{4}{169} \neq \frac{13}{169}$,we have $P(E \cap F) \neq P(E) \times P(F)$.
Therefore,the events $E$ and $F$ are not independent.