A simple acceptance-rejection method to simulate a vector (boldsymbol{X}) in the unit (d)-ball (left{boldsymbol{x} in mathbb{R}^{d} ight.)

A simple acceptance-rejection method to simulate a vector \(\boldsymbol{X}\) in the unit \(d\)-ball \(\left\{\boldsymbol{x} \in \mathbb{R}^{d}\right.\) : \(\|x\| \leqslant 1\}\) is to first generate \(\boldsymbol{X}\) uniformly in the hyper cube \([-1,1]^{d}\) and then to accept the point only if \(\|X\| \leqslant 1\). Determine an analytic expression for the probability of acceptance as a function of \(d\) and plot this for \(d=1, \ldots, 50\).