Note added: this seems related to Dirac's string trick!
This is a unclearly defined problem about a random walk, please excuse. I am looking for advice on how to make it well-defined.
Given is a sphere with radius R floating (at rest) in a liquid. A random walk through the liquid starts from the sphere surface until it gets to a given end point, say to distance of 100 R (in the same direction as the starting point, thus "above" the starting point).
The length L of the random walk is constrained and also given. Let us set it, for the sake of example, to L= 130 R.
Now, a number of these random walks will "go round" the sphere before they arrive at the end point. I have two questions: (1) What is probability for such a detour? (2) What is the probability for going "almost" round the sphere?
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Here is the original physical setting that I tried to translate into math. The problem is about a loose plastic chain attached to a plastic ball floating in a liquid, and the two questions are:
- What is the probability that the chain goes around the ball?
- What is the probability that the chain almost goes around the ball?
I am aware that "almost" must be defined better. I have no idea about how to do this. My friend and I are looking for advice.
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To clarify: this is not homework. It is a real discussion I am having with a friend. We are looking for ways to tackle the problem. We want to understand the probability that the plastic chains gets tangled up, depending on L. In fact, there are several such chains holding the ball, and we want to understand how often they will tangle, depending on L. But we worry that this problem in itself is much too complex.
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In Dirac's string trick, a ball attached by strings is rotated twice. After that, the strings are untangled while keeping the ball in place. The question asked here then becomes: What is the probability that the final untangling happens in the opposite direction? What is the probability that strings get tangled around the ball?
I found this video just now: in the video https://www.youtube.com/watch?v=XFLdq44aQvI the first 35 seconds show the process for which I try to know the probability.