Well it's an imaginative and soft question so take it as you wish .
Problem : imagine Spider-man walking on the cantor function (see the plot like the side of a building) :
Now he can throw a line to come faster to the top of the stairs . The problem is the spider-man webs can only solidify on an angle(i.e corner).
Question :
If spiderman have a limitied number of spider-webs which stay only on angle (corner) what is the probability he comes to the top of the stairs on a limited interval of the Cantor's function admitting it's his only means of transport ?
Do you have a strategy ?
I have no attempt because it's a new problem for me but Bernoulli's distribution could be a start .
Example of path to come on a horizontal line :
The new question is :
Finalizing Question :
If Spiderman is a point on that curve and have the probability $P=1/2$ to get a rational point $(x_i,y_i)$ on the curve he choose (The Cantor set here) and stop if the point is irrational what's the probability he goes at the top of the devil staircase ($x=y=1$) if he is assimilated to an increasing unknow $x$ starting to $x=0$ and $x_i<x_{i+1}$.In other word what's the probability he diverges (projective geometrical point of view) to $x=y=1$ choosing a number of rational point which is bounded ?
We can bound it with the Fréchet-Boole's inequality .
To figure it see :
Another one and Thomae function
The problem can be reformulate in term of Brownian motion :
Simplifying question for kids :
If now a kid plays with a Soap bubble and the bubble in a slight wind goes to a staircase what's chance the bubble don't collapse before the end of the next level of the home?
Reference :
https://en.m.wikipedia.org/wiki/Cantor_distribution
https://en.wikipedia.org/wiki/Fr%C3%A9chet_inequalities
https://en.wikipedia.org/wiki/Ford_circle
The basis of this answer and the strategy which come with it start :
What is the convex hull of the devil staircase taking only extremal point $P_i$ which is bounded ?
In fact if Spiderman (view as a point) have an illimited numbers of webs Jarvis march is sufficient if we admits there is a limited numbers of point in the convex hull .
Now as it's limited we can refine Jarvis march with Chan's algorithm which is more efficient .
There are other strategy but take it as an example for this problem of computational geometry .
Hope you find something clear and give your feedback.
Edit :
In term of probabilities see Expected number of vertices in a convex hull
From here (p.35) it seems that the probability is equal to $1/2$ is it right ?(false)
See https://people.maths.bris.ac.uk/~matmj/Demislides.pdf and Mahler question .
Using this paper the answer seems to be $P\simeq 1/3^n$ (false too)
I found an partial answer here Continued Fraction and Random Variable
And here "Probability" measures on Cantor set
Edit :
If we consider the term "equiprobable" so the answer is $1/2$ see
Thou shalt not say “at random” in vain: Bertrand’s paradox exposed
See theorem 5.1 Tchernoff bound here