**Toolbox:**

- P(C/B)=P\(\frac{(B\cap\;C)}{P(B)}\)
- P(\(\bar{A}\cap\bar{C}\))=P(\(\overline{A\cup\;C})\)
- =1-P(A\(\cup\)C)
- AlsoP(A\(\cup\)C)=P(A)+P(C)-P(A\(\cap\)C)

P(A)=\(\large\frac{2}{3}\)

P(B)=\(\large\frac{1}{3}\)

P(C)=\(\large\frac{1}{2}\)

P(A\(\cap\)C)=\(\large\frac{1}{5}\)

P(B\(\cap\)C)=\(\large\frac{1}{7}\)

P(C/B)=\(\large\frac{1}{7}\times\)\(\large\frac{3}{1}\)=\(\large\frac{3}{7}\)

P(A\(\cup\)C)=\(\large\frac{2}{3}\)+\(\large\frac{1}{2}\)-\(\large\frac{1}{5}\)

=\(\large\frac{29}{30}\)

P(A\(\cap\)C)=1-\(\large\frac{29}{30}\)=\(\large\frac{1}{30}\)