I came up with a random problem yesterday:
Suppose that in a random trial, each point $(x,y)$ where $x,y \in \mathbb{R}$ and $0 \leq x,y \leq 1$ is assigned a value of $0$ with 50% chance and a value of $1$ with 50% chance. What is a chance that there is a 'continuous path' that joins the point (0,0) and the point (1,1), such that along the path every single point has the same assigned value?
My intuition sort of says that the probability should be zero, but I don't really have a good reason why. For one thing, any particular path from (0,0) to (1,1) should have a 0% chance of occurring, but there are also an infinite number of possible paths from the point (0,0) to (1,1). In my head, the probability should depend on whether the "zero" was smaller than the "infinity"
$$P \simeq \text{num paths} * P(\text{chance a path happens})$$
To evaluate the number of total paths, I sort of imagined constructing paths with straight lines. To construct an arbitrary path between (0,0) and (1,1), I would pick some number of intermediate points inbetween the two corners, and specify their $x$ coordinate and $y$ coordinate. Something like:
$$\text{num paths} \simeq \aleph_1^2 + \aleph_1^4 + \cdot\cdot\cdot + \aleph_1^{\aleph_0} \simeq \aleph_1^{\aleph_0}$$
(First term is like I have to pick 2 real numbers for a single intermediate point, second term is if I pick 4 real numbers for two intermediate points, and so on)
To evaluate the chance that any particular path happens, it's sort of like saying that for every point along the line, it must be assigned the same value. Hence, its sort of like:
$$P(\text{chance a path happens}) \simeq \frac{1}{2}^{\aleph_1}$$
So combining the two, we have (very unrigorously):
$$P \simeq \aleph_1^{\aleph_0} * \frac{1}{2}^{\aleph_1}$$
This seems not only super unrigorous, but I'm not even sure if you can even do this kind of thing. My friend's words keep echoing in my head, "something something probability only means something when the sample space is actually defined properly something something" Is the problem (as I have posed it) even meaningful in any sense? And does my poor attempt at a solution have any grain of meaningfulness in it?