:: [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 cannot 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 cannot 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.)