Question 1201638
First let's solve for P[B] and P[D], since P[A] and P[C] are given.

P[A∪B]= P[A] + P[B] = 5/8, since A and B are disjoint.
      = 3/8 + P[B] = 5/8,  hence P[B] = 2/8 = 1/4

P[C∩D] = P[C]P[D] = 1/3, since C and D are independent.
       = 1/2 * P[D] = 1/3, hence P[D] = 2/3

a)  
P[A∩B] = 0, since A and B are disjoint.
P[B] = 1/4 (see above)
P[A∩Bc] = P[A] = 3/8, since A and B are disjoint, A is a subset of Bc.
P[A∪Bc] = P[Bc] = 1 - 1/4 = 3/4, since A and B are disjoint, A is a subset of Bc

b) 
A and B are not independent, since P[A∩B]=0 does not equal P[A]P[B] = 3/8 * 1/4.

c)
P[D] = 2/3 (see above)
P[C∩Dc] = 1/6, since P[C∩D] + P[C∩Dc] = P[C], i.e., 1/3 + P[C∩Dc] = 1/2.
P[Cc∩Dc] = P[(C∪D)c] 
         = 1 - P[C∪D] = 1 - (P[C] + P[D] - P[C∩D]) 
         = 1 - (1/2 + 2/3 - 1/3) =  1/6
P[C|D] = P[C] = 1/2, since C and D are independent.

d) C and Dc are independent, because P[C∩Dc] = P[C]P[Dc] = 1/2 * 1/3 = 1/6.