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

Each ticket is equally likely, so the supplied sums giveHenceRepeated numerical values still represent different equally likely tickets; they must retain their multiplicities.

Replacement and mixing make the one hundred draws independent and identically distributed. Writing $S={\sum }_{j=1}^{100}{X}_j$, linearity of expectation and additivity of variances of independent variables give

The event $S\ge 500$ is contained in $|S-400|\ge 100$. Thus Chebyshev's inequality givesThe variance bound is rigorous and uses no normal approximation.

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 $560$.