A moderator secretly adjusts a parameter $\theta>0$ of a machine that generates indepently random numbers from the intervall $[0,\theta]$. We produce $5$ random numbers with the machine and try to estimate $\theta$.
Now, our professor says that it is fair to assume that the generated random numbers obey a continuous uniform distribution, $\mathcal{U}_{[0,\theta]}$, and she sets up the statistical model $$ \left([0,\infty[^5,~~\mathcal{B}_{[0,\infty[}^5,~~\mathcal{U}_{[0,\theta]}^5\right). $$ However, I don't understand how this makes sense. If we evaluate the probability of a certain sample, say $(1,1,1,1,1)$ its probability is obviously $0$. If we use the probability measure from above it only provides the probability of getting a sample such that each number is $\leq 1$. So I dont quite understand how this probability measure can be used to estimate $\theta$?
What am I getting wrong?