Past exam of the options II course of the University of Oxford / 2012 / M / M4 / a / iv / Solution

The central limit theorem suggests approximating $S$ by a normal variable with mean $400$ and variance $560$. ThusSince $S$ is integer-valued, the usual continuity correction replaces $500$ by $499.5$:These are approximation formulas rather than consequences of Chebyshev's bound; the standard deviation is $\sqrt{560}$, not $560$.