what caused the confusion was your first example showed two jars X and Y which had just a few blue and orange fruits in them. So, you just close your eyes and pick a fruit. And you ask what the probability you chose a blue fruit from X.

The example you calculated after running a computer simulation 300 times of the same kind of choice, you got a square which showed you what fraction of times you got a particular choice. This said it was highly probable this gets you blue fruit from the X jar.

that what you are sampling in that case then is not the urns in your first example (there only 3 out of 27 blue fruits in X, so a random draw is not likely to get you one — -much less than 86% — - unlike your repeated simulation example.)

at least thats my impression. all your caluclations in the second case look correct and are straightforward (though if i recall alot of comments said somewhere in them originally you made some typos or simple arithmatic errors which you had fixed).
I’m really more or as much interested in the ‘philosophical’ differences or disputes between Byaesians and frequentists and my background was in stochastic processes and statistical mechanics so I never heard of Bayes theorem, though we used it — -it just wasn’t a named equation. Later i learned Jaynes redid stat mech based on Bayes theorem in 1950’s, and since then many physicists also call themselves Bayesians — -what physicists call maximum entropy approach they call a Bayesian approach. (I forget exactly but if i recall in statistical nonequilibrium physics, bayes theorem is called the K-L divergence though they are written very differently but mathematically equivalent.)

anyway, its possible i’m wrong, (as you said you chose a slightly more complex example — ie had a computer generated ‘ensemble’of many repeated experiments. thats what is done and useful for data analyses. i prefer absolute simplest examples. Alot of statistical physics uses ‘toy examples’ like flipping a coin or dice , doing a 1 dimesional random walk, etc.)