The article claims you can figure out the almost correct answer to the Monty Hall problem by running simulations hundreds of times instead of doing the maths (and the same for coin flips).

My problem is that it still relies on some mathematical intuition - that large sample sizes approximate the true distribution. Similarly bad intuition (like the gambler's fallacy) could easily be coded.

I agree that formally calculating the probabilities isn't necessary if you have the right intuition. But I believe getting good intuition is the result of training on problems (and then you can learn how to formalise it - which is the easier part).

Edit: Being good at mental arithmetic isn't necessary for programming, but being good at mental arithmetic isn't necessary for working as a mathematician either.