Question

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

Answer 1

`a.(1)/(13)\\b.(1)/(2)\\c.(1)/(13)\\d.(1)/(13)`

Explanation:

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

Total number of cards = 52

a)

Let A denotes the event that the card is a diamond​ and B denotes the card is red.

`P\left ( A|B \right )=(P\left ( A\cap B \right ))/(P(B))\\=((2)/(52))/((26)/(52))\\=(1)/(13)`

b)

Let A denotes the event that the card is red​ and B denotes the card is diamond.

`P\left ( A|B \right )=(P\left ( A\cap B \right ))/(P(B))=((2)/(52))/((4)/(52))\\=(1)/(2)`

c)

Let A denotes the event that the card is nine and B denotes the card is red.

`P\left ( A|B \right )=(P\left ( A\cap B \right ))/(P(B))=((2)/(52))/((26)/(52))\\=(1)/(13)`

d)

Let A denotes the event that the card is a king and B denotes the card is a face card.

`P\left ( A|B \right )=(P\left ( A\cap B \right ))/(P(B))=((4)/(52))/((12)/(52))\\=(1)/(13)`