# Use Chebyshev's Inequality to Find Upper Bound

The question I am looking over at right now is to use Chebyshev's inequality to compute an upper bound on the probability that my total earning is more than zero. The scenario is that I am playing a game (played $$n$$ times) where the chance of winning is $$p<\frac{1}{2}$$. If I win, I get a dollar, and if I lose, I have to pay a dollar.

Current Progress

I know that Chebyshev's inequality states that for $$X$$ as any random variable then for any $$t>0$$, then $$P(|X-E(X)|\geq t)\leq \frac{Var(x)}{t^2}$$. However, I am stuck as in finding the $$E(X)$$ and modifying the equation such that I can find the upper bound.

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$$\frac{X+n}2$$ has a binomial distribution with mean $$np$$ and variance $$np(1-p)$$
so $$\mathbb E[X] = n(2p-1)$$ and $$\text{Var}(X) = \frac14 np(1-p)$$