Probablity of normal distribution when x is a function

Question

Assume a uniform distribution random variable X~U(0,1). And $\Phi$ is the symbol of the standard normal distribution. Assume $Y=\Phi^{-1}(X)$. The question is, $\mathbb{P}(Y \le 0)=?$. The Solution is $1/2$, but I can't see why? I think $\Phi(x)$ is always positive?

Answer 1

It seems that $\Phi$ is meant to be the cumulative distribution function of the standard normal distribution. Since you conclude from $\Phi$ being positive that $\Phi^{-1}$ is positive, I get the impression that you're mistaking $\Phi^{-1}$ to refer to the reciprocal of $\Phi$. Here $\Phi^{-1}$ denotes the inverse of $\Phi$. Since $\Phi(0)=\frac12$, $\Phi^{-1}(x)$ is negative for $x\lt\frac12$ and positive for $x\gt\frac12$; hence the desired probability is $\frac12$.