Probability statistics problem

Please help with this problem

Given events A, B, and C, and P(A)=9/20, P(C)=3/8, P(A ∩ C)=2/10, P(B | A)=2/9, and P(B|Ac)=1/22

Note: Ac=A compliment

Find probability of the intersection of B and Ac

Find the probability of event B

Find the conditional probability of event A given event B

Supposed that event C was mutually exclusive of event B. Find the probability of event A ∪ B ∪ C

Thanks a bunch

1) By definition of conditional probability,
P(B|A^c) = P(B ∩ A^c)/P(A^c)
==> 1/22 = P(B ∩ A^c)/(1 - 9/20)
==> P(B ∩ A^c) = 1/40

2) Via total probability,
P(B) = P(B|A) P(A) + P(B|A^c) P(A^c)
.......= (2/9) * (9/20) + (1/22) * (1 - 9/20)
.......= 1/8.

3) By Baye's Law,
P(A|B) = P(B|A) P(A) / [P(B|A) P(A) + P(B|A^c) P(A^c)] = (2/9)(9/20) / (1/8) = 4/5.

4) P(A ∪ B ∪ C)
= P(A) + P(B) + P(C) - P(A ∩ B) - P(A ∩ C) - P(B ∩ C) - P(A ∩ B ∩ C)
= P(A) + P(B) + P(C) - P(A ∩ B) - P(A ∩ C) - 0 - 0, since C and B are mutually exclusive
= P(A) + P(B) + P(C) - P(B|A) P(A) - P(A ∩ C), via conditional probability
= 9/20 + 1/8 + 3/8 - (2/9) * (9/20) - 2/10
= 13/20.

I hope this helps!