No. Intention does matter. If the host chooses the goat by chance ( with the possibility to choose the car instead ) then my chances to win are 50% ( switching or not ).

But we are talking about Monty Hall after all. So I could be wrong :P

Nope. If the host shows the goat _for any reason_, then the 2/3 chance that the doors you didn't choose contain the car now applies to the one unchosen door remaining -- and you should switch.

Staying is an equally good option as switching in this version if the host revealed a goat, since the host randomly revealing a goat now makes it more likely that your initial choice was correct.

P(initial choice is correct) = 1/3

P(host shows goat) = 2/3

P(host shows goat | initial choice is correct) = 1

and vice versa P(host shows goat | initial choice is wrong) = 1/2

Applying Bayes:

P(initial choice is correct | host shows goat) = P(host shows goat | initial choice is correct) * P(initial choice is correct) / P(host shows goat) = 1 * 1/3 / (2/3) = 1/2

So, initial choice now has 1/2 odds for being right. The host revealing a goat gave us information on whether he was playing the P=1/2 or the P=1 game.

Intuitively applying this with the 100 door version: the host getting lucky and revealing 98 goats in a row makes it fairly likely that the initial door was correct all along and they didn't get super lucky avoiding the car 98 times, as there's a good chance the contestant helped them by hiding the car.

Perfect case study right here. It’s trivial to simulate and y’all aren’t budging. In this case, simulating, I see 2/3 for switching after the host shows a goat on purpose, and 1/2 for switching when the host happens to have shown a goat on accident.

It does matter if he meant to or not, because if he chooses randomly from the remaining doors, the probability of him revealing a goat depends on whether you have already chosen the car. Whereas if you know he could only ever reveal a goat, then the fact that he revealed a goat tells you nothing about your current selection.

If you are interested in more details, Wikipedia lists this as the "Ignorant Monty" variant.