Like André Nicolas, I do not understand your use of complex, especially with the $\ge$ symbol. So what follows ignores it. I will also use Capitals for random variables
$aX$ is a Gaussian random variable with mean $0$ and variance $a^2$.
Similarly $\sum_{i=1}^n b_i Y_i$ is a Gaussian random variable with mean $0$ and variance $\sum_{i=1}^n b_i^2$ because of independence.
And $aX - \sum_{i=1}^n b_i Y_i$ is a Gaussian random variable with mean $0$ and variance $a^2+\sum_{i=1}^n b_i^2$ because of independence.
So $\Pr( aX \geq \sum_{i=1}^n b_i Y_i ) = \frac{1}{2}$.