Three students $S_1, S_2$, and $S_3$ are given a problem to solve. Consider the following events: $U:$ At least one of $S_1, S_2$, and $S_3$ can solve the problem, $V: S_1$ can solve the problem, given that neither $S_2$ nor $S_3$ can solve the problem, $W: S_2$ can solve the problem and $S_3$ cannot solve the problem, $T: S_3$ can solve the problem. For any event $E$, let $P(E)$ denote the probability of $E$. If $P(U)=\frac{1}{2}, \quad P(V)=\frac{1}{10}, \quad$ and $\quad P(W)=\frac{1}{12}$ then $\mathrm{P}(\mathrm{T})$ is equal to