JEE Advance 2025 Paper 1

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 $P(U) = \frac{1}{2}, P(V) = \frac{1}{10}$  , and $P(W) = \frac{1}{12}$  , then $P(T)$   is equal to