Let [A] and [B] be two events such that [P(B A)=(2)/(5), P(A B)=(1)/(7)] and [P(A B)=(1)/(…

Question

Let $$\mathrm{A}$$ and $$\mathrm{B}$$ be two events such that $$P(B \mid A)=\frac{2}{5}, P(A \mid B)=\frac{1}{7}$$ and $$P(A \cap B)=\frac{1}{9} \cdot$$ Consider

(S1) $$P\left(A^{\prime} \cup B\right)=\frac{5}{6}$$,

(S2) $$P\left(A^{\prime} \cap B^{\prime}\right)=\frac{1}{18}$$

Then :

Accepted Answer

$$P(A/B) = {1 \over 7} \Rightarrow {{P(A \cap B)} \over {P(B)}} = {1 \over 7}$$

$$ \Rightarrow P(B) = {7 \over 9}$$

$$P(B/A) = {2 \over 5} \Rightarrow {{P(A \cap B)} \over {P(A)}} = {2 \over 5}$$

$$P(A) = {5 \over 2}\,.\,{1 \over 9} = {5 \over {18}}$$

$$S2:P(A' \cap B') = {1 \over {18}}$$

$S1:$ and $$P(A' \cup B) = {1 \over 9} + {6 \over 9} + {1 \over {18}} = {5 \over 6}.$$

Solutions · 1