Question

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.

Answer 1

To find the conditional probabilities, you need to use the definition of conditional probability. Given that a card is black, the probability that it is a spade is 1/2. Given that a card is a spade, the probability that it is black is 2. Given that a card is black, the probability that it is a seven is 1/13. Given that a card is a face card, the probability that it is a king is 1/3.

Step-by-step explanation:

To find these conditional probabilities, we need to use the definition of conditional probability:

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

a) The card is a spade, given that it is black:

In a standard deck of cards, there are 26 black cards and 13 spades. So, P(S|B) = P(S and B) / P(B) = 13/26 / 26/52 = 1/2

b) The card is black, given that it is a spade:

P(B|S) = P(B and S) / P(S) = 26/52 / 13/52 = 26/13 = 2

c) The card is a seven, given that it is black:

In a standard deck of cards, there are 4 black sevens and 26 black cards. So, P(7|B) = P(7 and B) / P(B) = 4/26 / 26/52 = 1/13

d) The card is a king, given that it is a face card:

In a standard deck of cards, there are 4 kings and 12 face cards. So, P(K|F) = P(K and F) / P(F) = 4/52 / 12/52 = 1/3

Answer 2

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.