Two buses ,4 and B are scheduled to arrive at a town central bus station at noon. The probability that bus A will
be late is l/5. The probability that bus B will be late is 7/25. The probability that the bus B is late given that bus
A is late is 9/10. Then,

$\begin{array}{l}\left(a\right) P\left(A\right)=\frac{1}{5},P\left(B\right)=\frac{7}{25},P(B/A)=\frac{9}{10}\\ \begin{array}{rl}P(\overline{A}\cap \overline{B})& =1-P(A\cup B)\\ & =1-\left[P\right(A)+P(B)-P(A\cap B\left)\right]\\ & =1-\left[\frac{1}{5}+\frac{7}{25}-P\left(A\right)\times P(B/A)\right]\\ & =1-\left[\frac{1}{5}+\frac{7}{25}-\frac{1}{5}\times \frac{7}{25}\right]=\frac{7}{10}\end{array}\end{array}$

$\begin{array}{r}\left(\mathrm{b}\right) \mathrm{P}(\mathrm{A}/\mathrm{B})=\frac{\mathrm{P}(\mathrm{A}\cap \mathrm{B})}{\mathrm{P}\left(\mathrm{B}\right)}=\frac{\mathrm{P}\left(\mathrm{A}\right)\mathrm{P}(\mathrm{B}/\mathrm{A})}{\mathrm{P}\left(\mathrm{B}\right)}\\ =\frac{\frac{1}{5}\times \frac{9}{10}}{\frac{7}{25}}=\frac{9}{50}\times \frac{25}{7}=\frac{9}{14}=\frac{18}{28}\end{array}$

$\begin{array}{c}\left(\mathrm{c}\right) \mathrm{P}(\mathrm{A}\cup \mathrm{B})=\mathrm{P}\left(\mathrm{A}\right)+\mathrm{P}\left(\mathrm{B}\right)-\mathrm{P}(\mathrm{A}\cap \mathrm{B})\\ =\mathrm{P}\left(\mathrm{A}\right)+\mathrm{P}\left(\mathrm{B}\right)-\mathrm{P}\left(\mathrm{A}\right)\mathrm{P}(\mathrm{B}/\mathrm{A})\\ =\frac{1}{5}+\frac{7}{25}-\frac{1}{5}\times \frac{9}{10}\\ =\frac{10+14-9}{50}\\ =\frac{3}{10}\end{array}$

$\begin{array}{r}\left(\mathrm{d}\right) \mathrm{P}\left({\mathrm{A}}^{\mathrm{\prime }}\cup {\mathrm{B}}^{\mathrm{\prime }}\right)=1-\mathrm{P}(\mathrm{A}\cap \mathrm{B})\\ =1-\mathrm{P}\left(\mathrm{A}\right)\mathrm{P}(\mathrm{B}/\mathrm{A})\\ =1-\frac{1}{5}\times \frac{9}{10}=\frac{41}{50}\end{array}$