PX262 - C1c - example problem

consider a quantum harmonic oscillator in the ground state to calculate probability of finding the particle displaced beyond its classical limit
working out the classical limit:

\begin{matrix} \frac{1}{2} M ω^{2} x_{m a x}^{2} = E_{0} = \frac{1}{2} ℏ ω \\ x_{m a x} = \sqrt{\frac{ℏ}{M ω}} = x_{0} \end{matrix}

the probability of the oscillator being beyond $| x_{0} | :$ $P = \int_{- \infty}^{x_{0}} | ϕ_{0} (x) |^{2} d x + \int_{x_{0}}^{\infty} | ϕ_{0} |^{2} d x$
the wavefunction: $ϕ_{0} = {(\frac{M ω}{ℏ π})}^{\frac{1}{4}} \exp (- \frac{M ω x^{2}}{2 ℏ}) = {(\frac{1}{π x_{0}})}^{\frac{1}{2}} \exp (- \frac{x^{2}}{2 x_{0}^{2}})$
let $t = \frac{x}{x_{0}} ⟹ d t = \frac{d x}{x_{0}} :$ $P = 2 \int_{1}^{\infty} \frac{1}{\sqrt{π}} e^{- t^{2}} d t = 0.045$
using $\int_{1}^{\infty} e^{- x^{2}} d x ≃ 0.16 \sqrt{2 π}$