Three students \(S_1, S_2\), and \(S_3\) are given a problem to solve. Consider the following events: \(U\) : At least o…

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

\(P(U)=\frac{1}{2},P(V)=\frac{1}{10},\text{and}P(W)=\frac{1}{12}\)

then \(P(T)\) is equal to

[JEE Advanced 2025]