We set, for all $t \in I$,
$$f ( t ) = \sum _ { n = 0 } ^ { + \infty } C _ { n } t ^ { n } \quad \text { and } \quad g ( t ) = 2 t f ( t ) .$$
Deduce that $\mathbb { P } ( T \neq 0 ) = 1 - \sqrt { 1 - 4 p ( 1 - p ) }$. Interpret this result when $p = \frac { 1 } { 2 }$.