stat problem

P(B)=0.6

P(A|B')=0.8

P(B|A')=0.7

how can I find the value of P(A) ?? THX

Sol

自行畫圖

令 P(AB)=x，P(A－B)=y，P(B－A)=0.6－x

0.8=P(A|B’)=P(AB’)/P(B’)=P(B－A)/(1－P(B))=y/0.4

y=0.32

0.7=P(B|A’)=P(A’B)/P(A’)=P(B－A)/(1－x－y)=(0.6－x)/(0.68－x)

0.476－0.7x=0.6－x

x=124/300

P(A)=x+y=124/300+0.32=11/15

Pr (B) = 0.6, hence, Pr (B') = 1 - Pr (B) = 0.4
Pr (A | B') = 0.8
Pr (B | A') = 0.7

Pr (B | A') = Pr (A' & B) / Pr (A')
Pr (B | A') x Pr (A') / Pr (B) = Pr (A' & B) / Pr (B) = [Pr (B) - Pr (A & B)] / Pr (B)
this is easily to be understand Pr (A' & B) = Pr (B) - Pr (A & B) with a Venn diagram

Hence,
Pr (A & B) = Pr (B) - Pr (A') x Pr (B | A') = Pr (B) - [1 - Pr (A)] x Pr (B | A')

By the law of total probability,
Pr (A) = Pr (A | B) x Pr (B) + Pr (A | B') x Pr (B') = {Pr (B) - [1 - Pr (A)] x Pr (B | A')} x Pr (B) + Pr (A | B') x [1 - Pr (B)] = [Pr (B)]2 - Pr (B | A') x Pr (B) + Pr (A | B') x [1 - Pr (B)] + Pr (A) x Pr (B | A') x Pr (B)

Hence,

P(A) = {[Pr (B)]2 - Pr (B | A') x Pr (B) + Pr (A | B') x [1 - Pr (B)]} / [Pr (B | A') x Pr (B)] = (0.62 - 0.7 x 0.6 + 0.8 x 0.4) / (1 - 0.7 x 0.6) = 0.26 / 0.58 = 13 / 29 = 0.4483

2010-03-02 14:34:33 補充：
brianwwc1993 is wrong because P(A and B) is not equal to P(B)xP(A|B')