\(P(A \cap \bar{B})=P(A)-P(A \cap B) \)
\(\frac{1}{2} =\frac{7}{10}-P(A \cap B) \)
\(P(A \cap B) =\frac{7}{10}-\frac{1}{2}=\frac{1}{5} \)

\(P(X / Y)=\frac{1}{2} \Rightarrow \frac{P(X \cap Y)}{P(Y)}=\frac{1}{2} \)
\(P(Y / X)=\frac{1}{3} \Rightarrow \frac{P(X \cap Y)}{P(X)}=\frac{1}{3} \)
\(\text {But } P(X \cap Y)=\frac{1}{6} \Rightarrow \frac{\frac{1}{6}}{P(Y)}=\frac{1}{2}\)
\(P(Y)=\frac{\frac{1}{6}}{\frac{1}{2}}=\frac{1}{3}\)
\(\text {and } P(X)=\frac{\frac{1}{6}}{\frac{1}{3}}=\frac{1}{2}\)
\(P(X \cup Y)=P(X)+P(Y)-P(X \cap Y)\)
\(=\frac{1}{2}+\frac{1}{3}-\frac{1}{6}=\frac{3+2-1}{6} =\frac{2}{3} \)

\(P(A / B)=\frac{P(A \cap B)}{P(B)}=\frac{P(A)}{P(B)}[\because A \subset B \Rightarrow A \cap B=A]\)

Fifty rupee note = 3
Hundred rupee note = 4
500 rupee note = 6
Totsl = 13
\(\mathrm{n}(\mathrm{S})=13 \mathrm{C}_{2}=78\)
\(\text {No. of ways the event to occur }=4\)
\(n(A)=4 C_{2}=6\)
\(\text {No. of ways the event not to occur }=72\)
The odds of event are a:b
\(6: 72 \Rightarrow 1: 12\)

\(P(\bar{A})=\frac{1}{4}, P(A) =\frac{3}{4}, P(A \cap B)=\frac{1}{4} \)
\(P(A \cup B) =\frac{3}{4}+P(B)-\frac{1}{4} \)
\(=\frac{1}{2}+P(B) \)
\(\frac{5}{6} =\frac{1}{2}+P(B) \)
\(P(B) =\frac{5}{6}-\frac{1}{2}=\frac{2}{6}=\frac{1}{3} \)
\(P(\bar{A})=\frac{1}{4}, P(A)=\frac{3}{4}, P(A \cap B)=\frac{1}{4} \)
\(P(A \cap B)=P(A) \cdot P(B)=\frac{3}{4} \times \frac{1}{3}=\frac{1}{4}=P(\text { Any }) \)
A and B are independent but not equally likely.