Lets say, I have the following situation:
I know, that an alarm will go on on a certain day. It will go on on any day from Monday to Sunday.
On the week before, I know that the possibility is equal (1/7).
My question: When it's Wednesday, two days have passed. Is the possibility for the alarm getting on still 1/7 or is it changed? And why or why not? I think the possibility didn't change, because it seems like to be like the monty hall problem.
This is not a Monty Hall situation, since the events of the first two days were not guaranteed to turn out that way – in the original Monty Hall the host-opened door is guaranteed to have a zonk. There is also no initial choice of which day is the special one (will have the alarm go off then), something which happens in Monty Hall with the initial choice of door by the player.
Accordingly, the probability that the alarm goes off on Wednesday given that the first two days had no alarm is raised to $\frac15$. Formally: $$P(\text{alarm goes on Wednesday}|\text{alarm doesn't go off on Monday or Tuesday})=\frac{1/7}{5/7}=\frac15$$