We consider $2n$ real random variables $c _ { 1 } , c _ { 2 } , \ldots , c _ { n }$ and $c _ { 1 } ^ { \prime } , c _ { 2 } ^ { \prime } , \ldots , c _ { n } ^ { \prime }$ that are mutually independent, all following the distribution $\mathcal { R }$.
Let $\left( \varepsilon _ { 1 } , \ldots , \varepsilon _ { n } \right) \in \{ - 1,1 \} ^ { n }$. Calculate $\mathbb { P } \left( \left( c _ { 1 } = \varepsilon _ { 1 } \right) \cap \cdots \cap \left( c _ { n } = \varepsilon _ { n } \right) \right)$.