How to identify an Independent Event

Question

In a well shuffled deck of 52 cards
E : The card drawn is black

F: The card drawn is King


Is above Event Independent or Not . Please explain me how to find it out ?

My Understanding   

Method 1 :

Independent events are events whose sample space of Happening will never overlaps

There for the there are 2 kings which are black in a deck so the above problem is NOT INDEPENDENT
 

Method 2

P( getting Black Card ) = 26/50  = 1/2

P( getting King ) = 4/52 = 1/13

P(Black and King)= 2/52 = 1/ 26

According to condition of independent Events
P( getting Black ) P( King ) = P( Black INSERSECTION King) = 1/2  * 1/13  = 1/26

Therefore the above problem IS INDEPENDENT

Which is Correct Approch ....? Pls Help me...

Answer 1

Method 2 is right.

Consider an example, of flipping two fair coins at two different places.

Bh : Head appeared on the coin flipped at Bangalore.

Bt : Tail appeared on the coin flipped at Bangalore.

Ph : Head appeared on the coin flipped at Pune.

Pt : Tail appeared on the coin flipped at Pune. 

As the both coins being used at Pune & Bangalore are fair, 

P(Bh) = P(Bt) = 1/2 and

P(Ph) = P(Pt) = 1/2.

The probability of getting head at both the places =

P(Bh ∩ Ph). 

But since the result of coin flip at on location is not going to affect the result at other location, they are independent.

so P(Bh ∩ Ph) = P(Bh)xP(Ph) = (1/2)x(1/2) = 1/4.

but here it can be observed drom the diagram below that, these two events (Bh & Ph) are having overlapping sample spaces:

Hence,   

"Independent events are events whose sample space of happening never overlaps."

This is not a correct argument.

Disjoint sample spaces of the two events tells us that the events are mutually exclusive.

And if any two events are mutually exclusive then they are highly dependent because occurrence of any one event immediately tells us the non occurrence of other event.

Also if A & B are two mutually exclusive events,

then A ∩ B = null.

so P(A ∩ B) will always be 0.

but if both P(A) & P(B) are non zero then P(A) x P(B) will never be equal to 0.

So A & B are not independent, if they are mutually exclusive and both of them having non zero probabilities.

Comments

Awesome :) Thank you