The difficulty of the Monty Hall Problem is not due to vague language (although some statements of it do suffer from vagueness). It is a genuinely astonishing result that many professional mathematicians, including statisticians, have tripped over. You should read about the story of Marilyn vos Savant and her columns discussing the Monty Hall Problem in Parade magazine, and the associated controversy. E.g. <https://chance.amstat.org/2022/11/monty-hall/>.

"vos Savant herself was flooded by a surge of disbelief. She received more than 10,000 letters from readers of her column, the vast majority of whom were absolutely convinced that she was wrong. Among them were many PhDs and a strikingly large number of mathematicians. One understanding mathematician kindly offered vos Savant some comforting words: “You made a mistake, but look at the positive side. If all those PhDs were wrong, the country would be in some very serious trouble.”"