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One card is drawn at random from a well shuffled deck of 52 cards. Are E and F independant if: E : the card drawn is a spade F : the card drawn is an ace

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

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

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