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We derive asymptotics for the probability that the origin is an extremal point of a random walk in ${ℝ}^n$. We show that in order for the probability to be roughly $1/2$, the number of steps of the random walk should be between ${\mathrm{e}}^{n/(Clogn)}$ and ${\mathrm{e}}^{Cnlogn}$ for some constant $Cgt;0$. As a result, we attain a bound for the $\frac{\pi }{2}$-covering time of a spherical Brownian motion.