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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 E and F independent? (i) E : the card drawn is a spade F : the card drawn is an ace (ii) E : the card drawn is black F : the card drawn is a king (iii) E : the card drawn is a king or queen F : the card drawn is a queen or jack.

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