**Toolbox:**

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

P(A)=\(\large\frac{1}{2}\) P(B)=\(\large\frac{1}{3}\)

\(P(A\;\cap\;B)\)=\(\large\frac{1}{4}\)

1=P(A/B)=\(\large\frac{1}{4}\times\)\(\large\frac{3}{1}\)

=\(\large\frac{3}{4}\)

2=P(B/A)=\(\large\frac{1}{4}\times\)\(\large\frac{2}{1}\)

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

3=\(P(\bar{A}\cap\;B\))=\(\large\frac{1}{3}\)-\(\large\frac{1}{4}\)=\(\large\frac{1}{12}\)

P(\(\bar{A}/B\))=\(\large\frac{1}{12}\times\)\(\large\frac{3}{1}\)=\(\large\frac{1}{4}\)

\(4=P(A\;\cup\;B)\)=\(\large\frac{1}{2}\)+\(\large\frac{1}{3}\)-\(\large\frac{1}{4}\)

\(P(\bar{A}\cap\;\bar{B})\)=\(1-\;P(A\;\cup\;B)\)

=1-\(\large\frac{7}{12}\)=\(\large\frac{5}{12}\)

P(\(\bar{A}/\bar{B}\))=\(\large\frac{5}{12}\times\)\(\large\frac{3}{2}\)

=\(\large\frac{15}{24}\)