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One card is drawn at random from a well shuffled deck of 52 cards. Are E and F independent if E : the card drawn is black F : the card drawn is a king?

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

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) = = $\large\frac{1}{13} + \frac{1}{13} = \frac{2}{13}$

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

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

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