Given events A B and C such that A and B are mutually exclus

Given events A, B and C such that A and B are mutually exclusive, A is a subset of C,P(B) = 1/2, P(C) = 3/4, P(B and C) = 3/8, and P(A and C) = 1/4, find:  
a. P(A)
b. P(C and not A)
c. P(B or C)
d. P(not A or B)
e. P(B|not C)

Solution

a)   P( A and C) = P(A)=1/4 as A is a subset of C

b) P(C and not A) = P(C) -P(A) = 3/4 -1/4 =1/2

c) P(B or C) = P(B) + P(C) -P(B and C) =1/2 +3/4 -3/8 =7/8

d) P(not A or B) = P(not A) + P(B) -P(not A and B) =(1-1/4) +1/2 -1/2 =3/4

e) P(B| not C) =P(B and not C) / P(not C) ={ P(B) -P(B and C)}/ (1-P(C))

                   = (1/2 - 3/8)/ 1/4 = 1/2