Independent card shuffle

Question

If a card is drawn at random from a well-shuffled deck, is getting a five independent of the card color (red or black)?

Answer 1

Let A be the event: the card chosen is red

Let B be the event: the card chosen is a 5

 

A and B are independent if P(A and B) = P(A)P(B)

 

P(A and B) = P(a red 5 was chosen) = 2/52 = 1/26

 

P(A)P(B) = (1/2)(4/52) = 1/26

 

Since P(A and B) = P(A)P(B), the events (A and B) are independent.  

 

Answer 2

Given that one did pick a 5, the probability of red 5 is ½. Color is independent because it's evenly distributed throughout the deck, regardless of 'rank' 2-10 or A, K, Q, J.