Let $X=(X_1,\dots,X_n)$ be a random element of $X_1\times\cdots\timesX_n$ with independent coordinates, and let $A$ be a measurable subset of the product space. Then \begin{align*} \mathbb E\left[\exp\left(\frac{1}{4}d_T(X,A)^2\right)\right] \le \frac{1}{\mathbb P(A)}. \end{align*} Consequently, for every $t\ge 0$, \begin{align*} \mathbb P\left(d_T(X,A)\ge t\right) \le \frac{1}{\mathbb P(A)}\exp\left(-\frac{t^2}{4}\right). \end{align*}