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.

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.