(i) E : the card drawn is black F : the card drawn is a king

In a deck there are 52 cards - 26 of each color and and there are 4 kings, one in each suit, and 2 of each color.

Given E: the card drawn is black, P(E) = $\large\frac{26}{52} = \frac{1}{2}$

Given F: the card drawn is a king, P(F) = $\large\frac{4}{52} = \frac{1}{13}$

If A and B are independent events, \(P(A\cap\;B)=P(A)\;P(B)\)

P (black card $\cap$ king ) = P (E $\cap$ F) = P (drawing an black king) = $\large\frac{2}{52} = \frac{1}{26}$

Now, P (E $\cap$ F) = P(E) $\times$ P(F) = $\large\frac{1}{2}$$\;\times$$\large\frac{1}{13} = \frac{1}{26}$

Therefore E and F are independent

(ii) E : the card drawn is a spade F : the card drawn is an ace

In a deck there are 52 cards - 13 of each suit - spades, clubs, diamonds and hearts, and there are 4 aces, one in each suit.

Given E: the card drawn is a spade, P(E) = $\large\frac{13}{52} = \frac{1}{4}$

Given F: the card drwan is an ace, P(F) = $\large\frac{4}{52} = \frac{1}{13}$

P (spade $\cap$ ace) = P (E $\cap$ F) = P (drawing an ace of spades) = $\large\frac{1}{52}$

P (E $\cap$ F) = P(E) $\times$ P(F) = $\large\frac{1}{4}$$\;\times$$\large\frac{1}{13} = \frac{1}{52}$

Therefore E and F are independent

(iii) E : the card drawn is a king or queen F : the card drawn is a queen or jack

Given a deck of 52 cards, there are 4 kings, 4 queens and 4 jacks, and the P(drawing a king) = P(drawin a queen) = P(drawing a jack) = $\large\frac{4}{52} = \frac{1}{13}$

Given E: the card drawn is a king or queen, P (E) = P(king) + P(queen) = $\large\frac{1}{13} + \frac{1}{13} = \frac{2}{13}$

Given F: the card drawn is a queen or a jack, P(F) = P(queen) + P(jack) = = $\large\frac{1}{13} + \frac{1}{13} = \frac{2}{13}$

P (E $\cap F) = P (card drawn is a queen) = $\large\frac{4}{52} = \frac{1}{13}$

P (E $\cap$ F) = P(E) $\times$ P(F) = $\large\frac{2}{13}$$\;\times$$\large\frac{2}{13} = \frac{4}{169} $$\;\neq \large\frac{1}{13}$

Therefore E and F are NOT independent