Independency of Disjoint Events

When we flip two fair coins simultaneously. Intuitively, the way one coin lands does not affect the way the other coin lands. The mathematical concept that captures this is called independence.

Definition of independent events : 
Events A and B are independent if –

 1. P(B)=0 or
 2. P(A | B) = P(A)

In other words, A and B are independent if knowing that B happens does not alter the probability that A happens, as is the case with flipping two coins.
Sometimes we get the idea that disjoint events are independent but in fact, the opposite is true.

Disjoint events are not independent.

Proof :
If two events are disjoint then we know that, A ∩ B = ∅ ; this means knowing that A happens means you know that B does not happen.
Now Assuming that,

P(B) != 0 .
P(A | B) = P( A ∩ B ) / P( B )

Since events are disjoint :

P( A ∩ B ) = 0
P(A | B) = 0 , which is not equal to P( A ).

Hence, the mathematical definition of independent events failed, and thus, it is proved that Disjoint events are not independent events.