I may be measuring the length of a stick. I then want to see what is the probability of one measurement under a model.
When using a continuous model, the probability of a single number is zero:
$$ \mathbb P( \text{Length}=\text{length} \mid \text{Model}=\text{model} ) = 0 \>. $$
What is right way to calculate the probability the measurement? Should I calculate the probability of an interval? If the precision of the ruler is 1 mm, should I use the $\text{length} \pm 0.5 \text{mm}$ interval, i.e.,
$$ \mathbb P( \text{Length} - 0.5 \text{ mm} \leq \text{Length} \leq \text{length} + 0.5 \text{ mm} \mid \text{Model}=\text{model} ) \>? $$