Events A and B are such that
P(A|B) = 0.3, P(A|B') = 0.8 and P(B) = 0.2
Work out P(B|A')

P(B|A') = 7/15

Step-by-step explanation:

P(B) = 0.2

P(A|B) = 0.3

P(A|B)=P(A and B) / P(B)

0.3=P(A and B) / 0.2

P(A and B)=0.06

P(B') = 1 - P(B) = 1 - 0.2 = 0.8

P(A|B') = 0.8

P(A|B')=P(A and B') / P(B')

0.8 = P(A and B') / 0.8

P(A and B') = 0.64

P(A)=P(A and B) + P(A and B')=0.06+0.64=0.7

P(A')=1-P(A)=1-0.7=0.3

P(A' and B) + P(A and B) = P(B)

P(A' and B) = P(B) - P(A and B)

P(A' and B) = 0.2 - 0.06 = 0.14

P(B|A') = P(A' and B) / P(A')

P(B|A') = 0.14 / 0.3 = 7/15

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4