I was asked the following question:
Someone gets into a clinic and sees that all of $2$ doctors treat patients.
The time of a treatment of all three people is an $\mathrm{IID}$ and has exponential distribution.
What is the probability that someone will get out last?
I tried to solve it but I want to make sure I'm on the right track.
$$\mathcal{P}(X_1>t,X_2\le t,X_3\le t)=\mathcal{P}(X_1>t)\cdot[1-\mathcal{P}(X_3>t)]\cdot[1-\mathcal{P}(X_2>t)]$$
I don't think the answer was meant to be in terms of $t$.
Well suppose you enter the clinic and see that there are two doctors who are treating patients right now.
Then you ask the question, what is the probability that I will be the last guy who leaves the hospital among the three patients ?
You know that the time of treatment of each patient is an exponential random variable. If you recall the exponential distribution has a 'Memorylessness' property - in other words the time until an event doesn't depend on how much time has already elapsed.
Now suppose that one of the patients is treated and you take his place. In this situation there are two patients each with treatment time which is exponential random variable. From the memorylessness property you and the other guy are equivalent as his waiting time 'resets' at each discrete moment. Then you have 50% chance to be the last guy who leaves the hospital (by symmetry).