Leave $ S_n $ be the sum of $ n $ i.i.d. copies of a real random variable $ X $. I'm interested in how big the amount is:

$$ alpha_X = Pr[|S_n|<1/R] cdot Pr[S_1 geq 1]$$

it can be, for parameters $ R $ Y $ n $.

Yes $ X $ It is a Rademacher or normal variable, then $ alpha_X = Theta (1 / R sqrt {n}) $. Yes $ X $ is the distribution that is $ 0 $ with probability $ 1-1 / n $ Y $ 1 $ with probability $ 1 / n $, so $ alpha_X = Theta (1 / n) $.

I would guess that $ alpha_X $ It can not be bigger than $ max (1 / R sqrt {n}, 1 / n) $. Is this correct?

by $ R = 1 $, the result is derived essentially from the local limit theorems. But I do not see how to argue in favor of big $ R $.