JEE Advanced 2025 Paper 1Question 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 $$P(U)=\frac12,\quad P(V)=\frac1{10},\quad \text{and}\quad P(W)=\frac1{12},$$ then $P(T)$ is equal to
Probability
Solution
Independent solving probabilities $a,b,c$: $V$ gives $a=\frac1{10}$. $P(U)=\frac12$ gives $(1-b)(1-c)=\frac59$; $W$: $b(1-c)=\frac1{12}$. Subtract: $1-c=\frac59+\frac1{12}=\frac{23}{36}$, so $P(T)=c=\frac{13}{36}$.