Fox 329

E[R] below can be computed depending on the random process... Try to guess what it is before computing. Randomness may be a fancy business...
See a similar concept: Bertrand Paradox

Dervish Fox: How can you cut a square randomly?

Red Fox: Here is a way; first generate a random number to represent point A. Then another random number for point B along the three others sides. Drawn the line AB, and cut it out.

Dervish: And how can you generate a random number?

Red: Excel does it easily. Type "=RAND()", and boom, you get one right away.

Dervish: But that depends on some seed numbers in the machine's memory, no?

Red: Yes, that's why they call them pseudo-random numbers. Do you have a problem with that?

Dervish: I have a bigger problem than that.

Red: So... you believe that random numbers can't exist.

Dervish: I believe that the word "random" should be removed from all world dictionaries.

Red: Radical.

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Fox 324

Before starting to publish received solutions for earlier foxes, let's start with a new one. Here, "random selection" means that points P and Q are uniformly distributed along 2 parallel sides.

Ahh, yes, expected value of the "ratio of" the bigger area to the smaller one is asked - obviously. Let's give some leeway to minor mistakes in our life. "Perfect" may sometimes be boring!

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Fox 201

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Fox 199

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Fox 198

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This is the beginning of a class of foxes with random processes.

The above process is a special form of a random walk on the plane.
We believe that this process is new! If you see something similar in literature, please let us know.

"Uniformly selected" means that every direction has an equal chance of being selected.
In other words: direction ~ Uniform[0, 360].
Point O is the origin. Generation of point A is irrelevant since OA does not change whether there will be a triangle or not at the end of the process.
Also, it is not important but assume that the random directions are generated clock wise.
The answer has not been confirmed yet, let us know if you find something very different.