Is there a simpler function with this shape?

If you need it to be smooth to a silly extent you can use the integral (or some approximation) of a bump function:

$$f(x) = \cases{\exp\left[-\frac{1}{1-x^2}\right] \hspace{1cm} x \in [-1,1]\\0 \hspace{3.cm} |x|>1}$$

This function is differentiable infinitely many times, and of course so will also it's integral be. But we will of course need to rescale it to fit the range for $x$: $[-1,1] \to [0,1]$ of course.

EDIT To answer to Bernards comment we may also need to renormalize it which we can do by dividing with $\int_{-\infty}^{\infty} f(x)dx$, that will ensure we get maximum of 1 (at $x=1$). Dividing by a constant won't change any of the other properties.

I suggest the following family of functions: $$f_a(x)=1-\sin\left(\frac{\pi}{2}(1-x)^2e^{-ax^2}\right).$$ Where you can choose $a$ as you like to satisfy your requirement. In the next image you see the cases $ a=1,10$.

Since you mentioned 'probability' have you considered a Weibull CDF?

$$ y = 1 - {e^{-(x/0.4)}}^{2.5} $$