Summary: A new upper bound on \(P(a_1\eta_1+a_2\eta_2+ \dots \geq x)\) is obtained, where \(\eta_1,\eta_2,\dots\) are independent zero-mean random variables such that \(|\eta|\leq 1\) for all \(i\). A multidimensional analogue of this result and extensions to (super)martingales are presented, as well as an application to self-normalized sums (or, equivalently, to \(t\)-statistics).