In a deck of $52$ cards , find the probability of drawing a king or a black card .

Let probability of drawing a black card be P(A)
Let probability of drawing a king be P(B)
As drawing king , drawing black card are not mutually exclusive ,
by "OR" I mean english word or in the following explanation.
I thought taking a king of spade or club will violate the condition black "OR" king thinking that when I take a king of spade or club will fulfil both the conditions. so I did as 
P(A OR B) = P(A) + P(B) - 2 P(A$\cap$B) 

= $ \frac{4}{52} + \frac{26}{52} - 2 \* \frac{2}{52}$

= $ \frac{26}{52}$

But the book told as per conventions as per the formula 
P(A OR B) = P(A) + P(B) - P(A$\cap$B) = $28/52$

Why is my reasoning wrong ?

when I draw a black spade or black club , won't I violate the condition that it is black "OR" king ? so only I subtracted possibilities of drawing black spade or black club . so I subtracted $2/52$ and got $26/52$ instead of $28/52$.

If I can also apply my reasoning as

P(black or king) = P(King that is not black) + P(Black card that is not a king)

= $2/52 + 24/52 = 26/52 $

Please help me in knowing where I am going wrong in reasoning 

So I am saying the formula to be used here is $P(\bar{A}\cap B) + P(\bar{B}\cap A)$ and book says the formula to be used here is $P(A \cup B)$. So in my logic for OR it is real A or B , there is not option for A and B.

My formula for for this question -> $P(\bar{A}\cap B) + P(\bar{B}\cap A)$ = $P(A) + P(B) -2\times P(A \cap B)$

Book's formula(conventional) for this question -> $P(A \cup B)$ = $P(A) + P(B) - 1 \times P(A \cap B)$