On the stage, next to the host, there are three doors (say Door number 1, Door number 2, and Door number 3). Behind one door, a car. Behind the other two doors, goats.

I am asked to pick a door, so I pick Door number 1.

Next, the host must follow the following rules:

1. Always open a door that was not picked by the contestant.

2. Always open a door to reveal a goat and never the car.

3. Always offer the chance to switch between the originally chosen door and the remaining closed door.

And he opens the Door number 2, which is a goat.

Then he asks me: will you switch your choice from Door 1 to Door 3?

I start to work on my strategy with following definitions:

A: the car is behind Door number 1

B: the car is behind Door number 2

C: the car is behind Door number 3

O₂: the host opened Door number 2

Based on statements and definitions above, find the following probabilities accordingly. (keep four decimal places)

P(O₂|A) =

P(O₂|B) =

P(O₂|C)=

P(C|O₂) =