I'm trying to solve following problem:
The symmetric coin is tossed 1600 times. What is the probability that the head will be shown up more than 1200 times?
As I know, I should use Chebyshev inequality, but I could not find mean and variance. Can you help me?
Consider random variable $X$ which is a number of times that head was show up. So, if we denote as $\xi_i$ the showing of head in $i$-th experiment, we will have
$$X = \sum_{i=1}^{1600}\xi_i.$$
The expectation is linear over them, so $$E(X) = E\left(\sum_{i=1}^{1600}\xi_i\right)=\sum_{i=1}^{1600}E(\xi_i) = \sum_{i=1}^{1600}\frac12 = 800.$$
All of the experiments are independent from each other, so the variance will be linear as well
$$\text{Var}(X) = \text{Var}\left(\sum_{i=1}^{1600}\xi_i\right)=\sum_{i=1}^{1600}\text{Var}(\xi_i) = \sum_{i=1}^{1600}\frac12\left(1-\frac12\right) = 400.$$
So, now you are able to apply Chebyshev inequality. Do you need further help with Chebyshev inequality?