if I m not clear enough please ask and I ll elaborate although I d rather discuss some actual changes rather than the problem solution which we both clearly understand well enough already the conditional probability zc in Chun can indeed be 1 2 in some cases 2 3 of them overall if the host prefers a certain goat door aka Monty Crawl in Rosenthal But in this situation the player can not gain or lose by switching Chun 1999 p 47 So it obviously doesn t change the player s strategy because in the remaining cases of Crawl which happen 1 3 of the time overall switching wins with probability zc 1 In more mechanistic terms if the host has a preference for a goat door there s a redistribution of the unconditoinal probability of winning z between two groups of cases that will have different zc but this redistribution is subject to the constraint that the sum is always 2 3 and the conditional probability in either group can not be less than 1 2 This is indeed fairly messy to explain in plain English and it s also not something that is assumed to be standard MHP by plenty of authors Rosenthal I can surely find more but surely others think this is what Savant meant for some reason There s also the question of whether the player knows what the host s strategy is I ve not seen Savant state that Since none of this actually affects the player s strategy for didactic reasons assuming the Wikipedia audience is a general audience I d prefer we leave the more complex discussion Monty Crawl Monty Small as a variant rather than making it the standard problem Perhaps we should mention in the lead that not assuming that Monty must randomly choose which door to open if both hide goats assumption changes the conditional probabilities for cases some go up some down but ultimately does not alter the player s optimal strategy because the conditional probabilities never drop below 1 2 for switching I deleted part of the aid to explanation section that actually solved the Monty Small problem to use Rosenthal s terms Using that as aid to understanding what the lead sets up as a simpler problem is a case of let me explain to you Galois theory so you better understand how to solve the quadratic equation Yes I m exaggerating a bit but I hope you get the point