Kolmogorov Inequality 
Let $X_1$ and $X_2$ be 2 independent random variables with expectations $\mu_1,\mu_2$ and variances $\sigma_1^{2},\sigma_2^{2}$ respectively. Let $\mu=\mu_1+\mu_2$ and $\sigma^{2}=\sigma_1^{2}+\sigma_2^{2}$. Show that 

$P[(|X_1-\mu_1|<t\sigma_1^{2})$ $\Lambda$ $(|X_1+X_2-\mu|<t\sigma^{2})]\ge1-t^{2}$ for every $t>0$. 
I think the question is wrong it must be $1-t^{-2}$. Is it so? Else give a hint to proceed.

You are correct that the statement is wrong. Take the simple case where $X_1$ is a standard normal and $X_2=0$, then the probability is
$$
P\left(|X_1-\mu_1|<t\,\sigma_1^{2}\ \Lambda\ |X_1+X_2-\mu|<t\,\sigma^{2}\right)
=P(-t<X_1<t).
$$
Ploting this against $1-t^2$ (red) and $1-{1\over t^2}$ (green) indicates you could be correct.

