Let A be the event that both of your two hole cards are face

Let A be the event that both of your two hole cards are face cards and let B be the event that both your hole cards form a pair. Are A and B independent?

Solution

A:both of your two hole cards are face cards
B:both your hole cards form a pair
P(A ) = 12C2/52C2 = 12*11/(52*51)
P(B) = 4c1*4c1/(52C2) = 16*2/(52*51)

P(A int B) = 4c1*4c1/(52C2) = 16*2/(52*51)

so,
P(A int B) is not equal to P(A)P(B) so they are not independent