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There isn't any sure-fire method of explaining anything, and especially in math. But specifically in the case of the Monty Hall problem it has been proven by extensive experience that many individuals with otherwise above average intellectual capacities exhibit an exceptional tenacity in refusing to accept the (otherwise) widely agreed upon solution; don't waste your time on repeating this experience, unless you should find pleasure in doing so.

It should also be noted that for experts in probability theory there is a lot of nitpicking to be done about the exact hypotheses that need to be formulated in order to get a completely well defined probabilistic problem with the purported solution as correct answer (the Wikipedia page on the subject illustrates this). So if the resistance you get is of the type: "it is not so clear cut, it might depend", then very possibly they could actually be right, unless you took all the required precautions in explaining the problem.

Supposing you are trying to convince one person (otherwise it may become complicated) who has honest misunderstanding (as opposed to for instance a problem of pride in admitting to have been wrong) I could suggest the following approach. First make sure you have the same understanding of the problem formulation; otherwise any reasoning is pointless. Then if the other person agrees this should be a well defined probability problem with an unambiguous answer (again pointless if not) let him/her propose a method that should be able to determine that answer; it does not have to be efficient, but it should be acceptable independent of your beliefs about the answer. Since you are (convinced to be) right, you can afford to be generous; of course you should take care to refuse obviously incorrect models (like anything where the winning location is determined/changes after the game starts). Some people would agree to be convinced by an experiment/simulation, but not everybody; you cannot impose this method. (Also if you do agree to this, be sure to fix beforehand a number of trials sufficient for you to accept the risk of being accidentally "shown" wrong; asking for more trials after such an eventuality would make your credibility zero.) Another option would be to make a tree of all possibilities, with agreed upon probabilities for descending into each branch. (Many arguments start "without loss of generality let the first door be chosen". Don't do such a thing; while it can be justified, you cannot expect your adversary to accept the WLOG). Then compute the probabilities for each leaf, and add up. Or some other method still that could be proposed. Just make sure you agree about the method and its validity before applying it. Then just work it out, and accept the result.