Question 855471
<pre>
 P(B)=0.2 P(A|B)=0.7 P(B')=0.8 AND P(A|B')=0.9 FIND P(B|A)

Let P(A&B')=w, P(A&B)=x, P(A'&B)=y, P(A'&B')=z, 

{{{drawing(300,200,-4,4,-2,4.8,rectangle(-4,-1.6,4,4.4), locate(-2,1.8,w),locate(1.5,1.7,y),locate(3.4,-1,z), locate(-3.6,2.5,A), locate(-.1,1.8,x),red(circle(-sqrt(2),sqrt(2),2)),red(circle(-sqrt(2),sqrt(2),1.95)),red(circle(-sqrt(2),sqrt(2),1.975)),
blue(circle(sqrt(2),sqrt(2),2),circle(sqrt(2),sqrt(2),1.95),circle(sqrt(2),sqrt(2),1.975)),
locate(3.4,2.5,B)
 )}}}

then 

(1)   w+x+y+z=1

(2)   P(B)= x+y = 0.2

(3)   P(A|B) = x/(x+y) = 0.7

(4)   P(B') = w+z = 0.8

(5)   P(A|B') = (w+x)/(w+z) = 0.9

From (1) and (2), 

(6)  w+z = 0.8 

From (2) and (3),

{{{system(x+y=0.2,x/(x+y)=0.7)}}} get 

(7)  x=0.14

(8)  y=.06

From (4) and (5)

(9)  w+x = 0.72

from (7) and (9)

(10) w = 0.58

From (6) and (10)

(11) z = 0.22 

{{{drawing(300,200,-4,4,-2,4.8,rectangle(-4,-1.6,4,4.4), locate(-3,1.8,w=0.58),locate(1.5,1.7,y=0.06),locate(2.3,-1,z=0.22), locate(-3.6,2.5,A), locate(-.2,2,"x="),locate(-.4,1.4,0.14),red(circle(-sqrt(2),sqrt(2),2)),red(circle(-sqrt(2),sqrt(2),1.95)),red(circle(-sqrt(2),sqrt(2),1.975)),
blue(circle(sqrt(2),sqrt(2),2),circle(sqrt(2),sqrt(2),1.95),circle(sqrt(2),sqrt(2),1.975)),
locate(3.4,2.5,B)
 )}}}

P(B|A) = x/(x+w) = 0.14/(0.14+0.58) = 0.14/0.72 = 14/72 = 7/36

Edwin</pre>