I am writing a problem such that calculus may not rigorously appear within it, and the usage of limits would be better avoided. Such constraints naturally cause complications. An altered version of the problem that retains the complication is as follows: A point P is randomly chosen from the interior of the unit circle centered at the origin. Let D equal the distance from this point to the point (0,2) to P. What is the value D will most likely take on?
The intended solution is to be $$\sqrt{5}$$ as the (infinite) set of points that are sqrt(5) away from (0,2) inside the circle lie on an arc which has greater length than the arcs that correspond to infinite sets of points that are other distances away from (0,2). However the probability that D equals sqrt(5) is 0, and there is a bijection from any arc (now taken as a set of points) to another (by dilation and translation) and so cardinality may be easily cited to claim that all values of D are equally likely. How might I specify that I really do want problem-solvers to compare the arcs not by their respective areas but by their lengths? I do not want to clutter the relatively simply worded problem by introducing rigorous definitions (ie a limit definition that gives arcs infinitesimal widths) and would rather include a brief disclaimer within the problem. An ideal solution would be to ask for a quantity D* that is briefly defined in such a way that no misconception can occur.
Thanks in advance!
If I understand what you're asking for correctly, I believe you want the $x$ value (with the $x$-axis corresponding to all possible distances $D$) where the maximum $y$ value occurs in the Continuous probability distribution graph.
The $x$-axis for your question is, in general, from $-\infty$ to $\infty$, but for distances this can be limited to $0$ to $\infty$ since distances are non-negative. However, in your specific case, you only need to be concerned with the range of allowable distances of $1$ to $3$, so if the $x$-axis extends outside this range, the $y$ value will just always be $0$ there. Note the total area under this type of graph is always $1$. As stated in the link, the $y$ value is the Probability density function, with this being an appropriately normalized value of the arc length (the normalization is such to ensure the total area under the graph is $1$) you mentioned in your question. Also, as indicated in the first link, the probability of the value of the distance $D$ being between $D_1$ and $D_2$ would be the area under the graph between those $2$ points on the $x$-axis.
If this is not what you're looking for, please give some more details. Thanks.