Is this a modified Monty Hall problem (numbered doors)?

The interviewer is wrong in stating that Monty "clearly" picked door 3 and that therefore the chances are 50-50. Monty picked door number 3 in this particular case but he didn't say why.

The interviewer is correct that if we do not know why Monty showed door number 3 (does he always show door number 3 no matter what is behind it or what you pick-- does he always pick a door at random that you did not pick no matter what is behind it-- does he always show a door with a goat you did not pick?) then the question can not really be answered. (But he is wrong in assuming that it is 50-50). We have to make an assumption but... what assumptions are valid and which aren't?

Here are several possible rules Monty could be played by:

Classic: Monty always shows you a goat you do not pick. Strategy: switch. 2 out of 3 in your favor.

Random: Monty picks a door you did not pick and this time it just randomly happened to be a goat. Strategy: doesn't matter. 2 out of 4 whether you switch or stay.

Devious: If you pick the car Monty will show you a goat in the hopes that you will assume a classic game. If you pick a goat he won't give you a choice. Strategy: stay. 100% in your favor.

Tough luck: Monty will always show you the car if he can. He'll only show you a goat if you pick the car. Strategy: stay. 100% in your favor.

Warped: There is one goat with a spotted tail. Monty will always show you a door you did not pick that does not have the spotted tail goat. Strategy: stay. If you switch it is 2 in 3 that you will get the goat with the spotted tail. So it is 2 in 3 if you don't switch you get the car.

Hierarchical: If you pick the goat with spotted tail, Monty will show you the goat without the spotted tail. If you pick the goat without the spotted tail Monty will show you the car. If you pick the car Monty will show you the goat without the spotted tail. Strategy: 50-50.

etc.

Which is the more likely one he is playing? We can't tell. And obviously these are not the only strategies.

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Actually what would be fair is if it were worded like this:

You are an a game show and where you have a chance to pick a car or two goats. The hosts goal is to give you a goat and keep you from picking a car. You pick door 1 and he shows you door 3 has a goat and offers you a chance to switch to door 2. Should you?

Answer: it doesn't matter. Whichever door you pick he will put the goat behind it after you pick it.

I'd say the interviewer was at least misleading.

the rules need to be made clear. the standard rule is: Monty opens a worthless door from the two you didn't choose, in the scenario that both of the unchosen doors are worthless, he chooses between them with equal probability.

Perhaps your interviewer had different rules in mind. It's different, for example, if Monty himself has no idea which doors are worthless. It's different if he always opens door $3$ regardless of its contents (though what does he do if you have chosen door $3$?).

In any case, if the rules are not clear then there is no way to answer the question.

Here is what I believe is happening.

In the classical MH problem, you pick a door, let's say door #1, and then Monty opens some other door which definitely has a goat behind it. This table from the Wiki page, which assumes you picked door #1 and that Monty always reveals a goat, summarizes the scenarios very nicely:

\begin{array}{|c|c|c|c|c|} \hline \textbf{Behind door 1} & \textbf{Behind door 2} & \textbf{Behind door 3} & \textbf{Result if stay} & \textbf{Result if switch}\\ \hline \text{Car} & \text{Goat} & \text{Goat} & \text{Win} & \text{Lose}\\ \hline \text{Goat} & \text{Car} & \text{Goat} & \text{Lose} & \text{Win}\\ \hline \text{Goat} & \text{Goat} & \text{Car} & \text{Lose} & \text{Win}\\ \hline \end{array}

Suppose instead that you choose door #1 and Monty always picks door #3, regardless of what's behind door #3. If door #3 opens and you see the car, that means you lose. But that's not the scenario the interviewer gave. The interviewer said the door was opened and a goat was revealed. That means that the only applicable scenarios from the table above are rows 1 and 2 (where row 1 is the first row after the header row), because only those rows correspond to a goat being behind door #3. Based on that, do you see how it becomes 50/50?

If this is really what the interviewer intended then I do believe that the interviewer should've been more clear that Monty always opens door #3 regardless of what it's hiding.