I’m sure it was just the Maths Professors equivalent of pulling wings of a fly. Taking pleasure in making perspective students squirm after a long year teaching their forerunners
Yes, probably, but presumably a plan which will have 'failed' if/when they came across an applicant who was already familiar with the problem'/paradox in question!
For Monty Hall, I mapped out the permutations and calculated the probabilities. Once I could see the permutations it made sense.
I suppose that's essentially what I did when I eventually 'resorted to simulation', since the maths (and correct mathematical answer) then 'made sense'. However, despite that, I have to say that, even decades on, I still have difficulty in 'getting my head around it', since I find it difficult to 'conceptualise' the situation (non-mathematically) in a manner which does not appear to violate the most basic concepts of probability theory!
Birthday paradox is more abstract, to my mind at least
I certainly find it more interesting/useful, and it's a bit different (from Monty Hall), since it's really just a matter of 'surprise' ('incredulity'?!) regarding the mathematical provable answer, rather than any apparent 'conceptual violations'.
I find it 'useful' because, just as my introducing it into this thread, it is a concept that can be extended into many real-world situations. If one has three items, each with the same (very small) risk of failure in any week in the next five years, then the probability of them all failing in the same week is very much higher than the "astronomically small probability" that many/most of us would probably initially think was the 'intuitive' answer (as with the birthday paradox).
It’s odd because I can visualise some exponential compounding expansions. For example collision theory in cryptography I have less trouble visualising. √n is ~ number required for a 50% chance of a match with n items for example. Which of course is another approximation for the Birthday Paradox.
Even mathematicians start life as human beings with intuition, and I'm sure that, in many/most cases such as we are discussing those 'initial intuitions' prove to have been wildly 'off' once one has done, and seen the results of 'the sums'!
Whether in relation to 'doubling up' of grains of sand across the squares of a chess board, or a single bacterium 'doubling' every 20 minutes for 24 hours, I don't think that the 'intuitions' of many people who have not 'done such sums' (or seen the results thereof) will be even remotely expecting the 'astronomical' numbers which result - and, to be topical, I imagine much is the same in terms of many people's understanding (or lack of it) of the evolution of a viral epidemic/pandemic!
Kind Regards, John