Suppose there are three students wanting to meet a Professor in his office. The time $i$-th student takes with the Professor is $Exponential(\lambda_i)$ distributed, and they are independent. Assume all students start talking to the Professor simultaneously. Find the distribution of the time until only one student is left.
If I denote $T$ to be the desired time and $T_i$ to be the discussion time for $i$-th student, how exactly can I write $\{T\leq x\}$ in terms of $T_i$?
It seems that $\{T\leq x\}=\{T_1\leq x,T_2\leq x\}\cup\{T_2\leq x,T_3\leq x\}\cup\{T_1\leq x,T_3\leq x\}$.
Am I right?
Your expression is right. To evaluate the probability, we use Inclusion/Exclusion, getting $$\Pr(T_2\le x)\Pr(T_3\le x)+\Pr(T_1\le x)\Pr(T_3\le x)+\Pr(T_1\le x) \Pr(T_2\le x)\\-2\Pr(T_1\le x)\Pr(T_2\le x)\Pr(T_3\le x).$$