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I have been trying my best at figuring out what is really going on in the problem introduced in the video: https://www.youtube.com/watch?v=RuSdSgbQQWc

So far, i have only managed to show the few results he takes for granted (that the radius of the inscribed circle is half of the outer circle and so on). One thing i did notice about argument nr. 3 is that the center is the midpoint of an infinite amount of chords (by rotating the diameter about the midpoint), and so, a given midpoint may contribute with more than one chord, and therefore also a greater probability of occuring. There is also something fishy about how the chords are chosen i argument nr. 2 from the midpoints, but i can't put my finger on what...

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    Quote from the YouTube page: "I go through the steps slowly to try and unravel one of mathematics greatest mystery. A geometric probability paradox that defies all logic and reasoning. Beware, this might just destroy your mind." I stopped there. – 2017-01-24
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    I know, it is very cringeworthy, but it is actually not as trivial to explain where his arguments go wrong as you might be expecting. – 2017-01-24
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    This is the Bertrand Paradox. For more explanation, see https://en.wikipedia.org/wiki/Bertrand_paradox_(probability) and http://math.stackexchange.com/questions/1512992/bertrand-paradox-solutions – 2017-01-24

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The root of the problem is that the term "randomly choosing a chord" is undefined.

Probability theory works with models. The moral of the Bertrand example is that we work with models and that it is not paradoxical that calculations based on different models result in different numbers. It also demonstrates that our intuition does not work very well when it comes to probability. Interestingly, nobody is so surprised if something goes wrong in astronomy if we do not use general relativity... (Recently, Samsung specialists used a wrong model when calculating the thermodynamics of the battery they used in their smart phones. Not a paradox, "only" an accident.)

When I first encountered the Bertrand paradox, I planned a physical experiment because I wanted to see which model (of the usual Bertrand varieties) would match my experiment.

Here is the description of the experiment:

Take a large sheet of paper. Draw circles of diameter of $4$ cm on the paper. (It will be clear soon why $4$ cm.) Place the paper on the floor. Then take a plastic ruler. Throw it up so it bounces back from the sealing. If the ruler falls on the paper then draw a line on the paper and lengthen the line if necessary. Then measure the lengths of the chords cut by the line. (If no circle intersected then do nothing.) The figure below depicts one successful experiment resulting in $4$ chord measurements:

enter image description here

I collected $3000$ "random" chord lengths and calculated the average. The average turned out to be near to $\pi$, it was actually $3.146$! So much for physical reality.

Now if one uses the following model for the random choices then the model will describe/explain quite well the result of the experiment above.

The model

Take a circle of diameter $D$. Select a random point on the diameter. (Use the uniform distribution on $[0,D]$.) At the point selected randomly erect a perpendicular line determining a chord. Compute the mean of the random chord length. The mean turns out to be

$$\frac{D\pi}4.$$

So much for the model.

This is why I chose $4$...

Would it have been paradoxical if I had not been found an average near to $\pi$? No. It simply would have shown that I chose the wrong model.