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A card is drawn at random from a standard deck of 52 cards. Find the following conditional probabilities. ​a) The card is a spade​, given that it is black. ​b) The card is black​, given that it is a spade. ​c) The card is a seven​, given that it is black. ​d) The card is a king​, given that it is a face card.

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LammettHash

The probability of an event A occurring given that B has occurred is

P(A | B) = P(A and B) / P(B)

a. By the definition above,

P(spade | black) = P(spade and black) / P(black)

  • P(black) = 26/52 = 1/2 because 26 of the 52 cards have a black suit
  • All spade cards are black, so P(spade and black) = P(spade) = 13/52 = 1/4

Then P(spade | black) = (1/4) / (1/2) = 1/8.

b. We can do the same breakdown as in (a), or we can make use of the definition of conditional probability

P(A | B) = P(A and B) / P(B) = (P(B | A) * P(A)) / P(B)

Then

P(black | spade) = (P(spade | black) * P(black)) / P(spade)

  • P(black) = 1/2
  • P(spade) = 1/4
  • P(spade | black) = 1/8

Then P(black | spade) = (1/8 * 1/2) / (1/4) = 1/64.

c. By definition,

P(7 | black) = P(7 and black) / P(black)

  • P(7 and black) = 1/52 because this is a unique card
  • P(black) = 1/2

Then P(7 | black) = (1/52) / (1/2) = 1/104.

d. By definition,

P(king | face) = P(king and face) / P(face)

  • All kings are face cards, so P(king and face) = P(king) = 4/52 = 1/13
  • P(face) = 12/52 = 3/13

Then P(king | face) = (1/13) / (3/13) = 1/3.

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