Really Hard Conditional Probability Question

given
P(A|B) = 0.82
P(A'|B') = 0.82
P(B) = 0.81

Find P(B'|A). Show working please.

P(A/B') + P(A'/B') = 1 So P(A/B') = 1-0.82 = 0.18

P(B') = 1- P(B) = 0.19

P(B'/A) = P(A/B')P(B') /[P(A/B)P(B) +P(A/B')P(B')] = 0.18*0.19/[0.82*0.81 + 0.18*0.19]
=0.0489691

labelling:
w=P(AB)
x=P(AB')
y=P(A'B)
z=P(A'B')

a=P(A|B)=P(AB)/P(B)=w/(w+y)
b=P(A'|B')=P(A'B')/P(B')=z/(x+z)
c=P(B)=(w+y)
d=P(B'|A)=x/(w+x)
w+x+y+z=1

w
=(w/(w+y))(w+y)
=a*c

c=w+y
y
=c-w
=c-(a*c)
=c(1-a)

z/(x+z)=b
z=bx+bz

w+x+y+z=1
ac+x+c(1-a)+z=1
x+z+c=1

bx+(b-1)z=0
x+z=1-c
bx+bz=b(1-c) //subtract 1st from 3rd
z=b(1-c)

x=(1-c)-b(1-c)
x=(1-b)(1-c)

d=x/(w+x)
d=(1-b)(1-c)/(ac-(1-b)(1-c))

P(A'|B) = .18
P(A|B') = .18
P(B') = .19

P(AB) = .81 * .82 = .6642
P(A'B') = .82 * .19 = .1558
P(A'B) = .18 * .81 = .1458
P(AB') = .18 * .19 = .0342

So P(B'|A) = .0342 / (.6642 + .0342) = 0.0490