= Solution
For the <simple random walk on the integer line>, a return to the origin is possible only at an <even number>[even] time. At time $2n$, exactly $n$ of the increments must be $+1$, so the return probability is
$$
p_{2n}=\frac1{2^{2n}}\binom{2n}{n}.
$$
The supplied factorial bounds, equivalently the order estimate in the <Stirling formula>, give
$$
p_{2n}\asymp n^{-1/2}.
$$
Consequently $\sum_np_{2n}$ diverges. By the <recurrence criterion by return probabilities>, the origin is a <recurrent state>; translation invariance then makes the whole walk recurrent.
For three <independent random variables>[independent] walks, the probability that all three are at the origin at time $2n$ is
$$
p_{2n}^3
=\left(\frac{\binom{2n}{n}}{4^n}\right)^3
\asymp n^{-3/2}.
$$
This <P-series> is convergent, so the first of the <Borel-Cantelli lemmas> says that simultaneous returns occur only finitely often with probability one. Therefore the requested probability is
$$
\boxed{0}.
$$
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