10. If A and B are two independent events, prove that $\mathrm { P } ( \mathrm { A } \cup \mathrm { B } ) . \mathrm { P } \left( \mathrm { A } ^ { \prime } \cap \mathrm { B } ^ { \prime } \right) \leq \mathrm { P } ( \mathrm { C } )$, where C is an event defined that exactly one of A and B occurs. Sol. $P ( A \cup B ) . P \left( A ^ { \prime } \right) P \left( B ^ { \prime } \right) \leq ( P ( A ) + P ( B ) ) P \left( A ^ { \prime } \right) P \left( B ^ { \prime } \right)$ $= \mathrm { P } ( \mathrm { A } ) \cdot \mathrm { P } \left( \mathrm { A } ^ { \prime } \right) \mathrm { P } \left( \mathrm { B } ^ { \prime } \right) + \mathrm { P } ( \mathrm { B } ) \mathrm { P } \left( \mathrm { A } ^ { \prime } \right) \mathrm { P } \left( \mathrm { B } ^ { \prime } \right)$ $= \mathrm { P } ( \mathrm { A } ) \mathrm { P } \left( \mathrm { B } ^ { \prime } \right) ( 1 - \mathrm { P } ( \mathrm { A } ) ) + \mathrm { P } ( \mathrm { B } ) \mathrm { P } \left( \mathrm { A } ^ { \prime } \right) ( 1 - \mathrm { P } ( \mathrm { B } ) )$ $\leq \mathrm { P } ( \mathrm { A } ) \mathrm { P } \left( \mathrm { B } ^ { \prime } \right) + \mathrm { P } ( \mathrm { B } ) \mathrm { P } \left( \mathrm { A } ^ { \prime } \right) = \mathrm { P } ( \mathrm { C } )$.
If $\theta$ and $\phi$ are acute angles satisfying $\sin \theta = \frac { 1 } { 2 }$ and $\cos \phi = \frac { 1 } { 3 }$, then $\theta + \phi$ belongs to
10. If A and B are two independent events, prove that $\mathrm { P } ( \mathrm { A } \cup \mathrm { B } ) . \mathrm { P } \left( \mathrm { A } ^ { \prime } \cap \mathrm { B } ^ { \prime } \right) \leq \mathrm { P } ( \mathrm { C } )$, where C is an event defined that exactly one of A and B occurs.
Sol. $P ( A \cup B ) . P \left( A ^ { \prime } \right) P \left( B ^ { \prime } \right) \leq ( P ( A ) + P ( B ) ) P \left( A ^ { \prime } \right) P \left( B ^ { \prime } \right)$\\
$= \mathrm { P } ( \mathrm { A } ) \cdot \mathrm { P } \left( \mathrm { A } ^ { \prime } \right) \mathrm { P } \left( \mathrm { B } ^ { \prime } \right) + \mathrm { P } ( \mathrm { B } ) \mathrm { P } \left( \mathrm { A } ^ { \prime } \right) \mathrm { P } \left( \mathrm { B } ^ { \prime } \right)$\\
$= \mathrm { P } ( \mathrm { A } ) \mathrm { P } \left( \mathrm { B } ^ { \prime } \right) ( 1 - \mathrm { P } ( \mathrm { A } ) ) + \mathrm { P } ( \mathrm { B } ) \mathrm { P } \left( \mathrm { A } ^ { \prime } \right) ( 1 - \mathrm { P } ( \mathrm { B } ) )$\\
$\leq \mathrm { P } ( \mathrm { A } ) \mathrm { P } \left( \mathrm { B } ^ { \prime } \right) + \mathrm { P } ( \mathrm { B } ) \mathrm { P } \left( \mathrm { A } ^ { \prime } \right) = \mathrm { P } ( \mathrm { C } )$.\\