Acard 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.

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)

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)

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

c. By definition,

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

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

d. By definition,

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

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