matrix matching type

A) $P(A\cap B)=P\left(A\right)+P\left(B\right)-P(A\cup B)$

$=\frac{3}{7}+\frac{4}{7}-\frac{5}{7}$

$=\frac{7-5}{7}$

$=\frac{2}{7}$

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

$=\frac{\frac{2}{7}}{\frac{4}{7}-\frac{2}{7}}=\frac{\frac{2}{7}}{\frac{2}{7}}=1$

B) $E_1-$ The letter has came TATANAGAR

$E_2-$ The letter has came CALCUTTA

Let A be first two consecutive letters on envelope are TA

$P\left(E_1\right)=\frac{1}{2},P\left(E_2\right)=\frac{1}{2}$

$\begin{array}{l}P\left(\frac{A}{E_1}\right)=\frac{1}{7},P\left(\frac{A}{E_2}\right)=\frac{2}{8}\\ P\left(\frac{E_2}{A}\right)=\frac{P\left(A\right)P\left(\frac{A}{E_2}\right)}{P\left(A\right)P\left(\frac{A}{E_1}\right)+P\left(A\right)P\left(\frac{A}{E_2}\right)}\end{array}$

$=\frac{\frac{1}{2}\left(\frac{2}{8}\right)}{\frac{1}{2}\cdot \frac{1}{7}+\frac{1}{2}\cdot \frac{1}{8}}$

$=\frac{\frac{11}{2}\left(\frac{2}{8}\right)}{\left(\frac{1}{7}+\frac{2}{8}\right)}$

$=\frac{\frac{2}{8}}{\frac{8+14}{7\times 8}}$

$=\frac{14}{22}=\frac{7}{11}$

Required probability $1-P\left(\frac{{\displaystyle E_2}}{{\displaystyle A}}\right)=1-\frac{7}{11}=\frac{4}{11}$.

C) $S=\{MM,FM,MF,FF\}$

Let A be the even that almost one of them is female

$A=\{MM,MF,FM\}$

Let B be the event that both are male

$B=\left\{MM\right\}$

$A\cap B=\left\{MM\right\}$