Past exam of the mathematics course of the University of Cambridge / 2008 / ia / Paper 2 / 4F / Solution

Let $X$ be one score. It has the discrete uniform distribution on $1,\dots ,6$, so its expected value and second moment areThereforeFor independent throws, linearity of expectation and additivity of variance give $\mathbb{E}{Y}_n=7n/2$ and $\mathrm{Var}(Y_n)=35n/12$. The sample average consequently has expectation $7/2$ and variance $35/(12n)$.

The Chebyshev inequality follows by applying the Markov inequality to the nonnegative square of a centered random variable. It givesThis is at most $1/10$ if $n\ge 350/27$. Thus $n=13$ suffices, as does every larger integer. This is the smallest integer guaranteed by this particular Chebyshev inequality bound; it need not be the smallest integer satisfying the actual probability requirement.