Independent events means P(A ∩ B) = P(A) x P(B)

P(A' ∩ B') = 1 - P(A ∪ B) = 0.15

So P(A ∪ B) = 1 - 0.15 = 0.85


But P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

P(A ∪ B) = P(A) + P(B) - P(A) x P(B) since they are independent

So 0.85 = 0.75 + P(B) - 0.75 P(B)

0.1 = 0.25 P(B)

P(B) = 0.1/0.25 = 0.4

P(A ∩ B) = P(A) + P(B) - P(A ∪ B)

= 0.75 + 0.4 - 0.85
= 0.3

From info,

P(C) ≠ 0

B and C are mutually exclusive, so this means P(B ∩ C) = 0 (i.e B and C don't intersect)

Since C is a subset of A, C could be
① smaller than A and inside A
or
② equal to A.

So we would think at first P(C) that ≤ 0.75

But, B and C are mutually exclusive. So this means that :

The portion where A ∩ B does not intersect C for this to be true.

So ② is not possible.

So we find the part of A that does not intersect B. That is where C can be in.

P(B' ∩ A) = P(A ∪ B) - P(B)
= 0.85 - 0.4
= 0.45

Or

P(B' ∩ A) = P(A) - P(A ∩ B)
= 0.75 - 0.3
= 0.45

Since P(C) ≠ 0,

Then we can conclude that :

0 < P(C) ≤ 0.45