The Monty Hall Paradox

Consider the following game.
There are $3$ doors and a prize is hidden at random behind one of them. A contestant chooses a door, say door $1$ for example, but does not open it. Let $D_i$ be the event "the prize is behind door $i$", for $i = 1,2,3$.

1. What is the probability $P(D_i)$ of the event $D_{i}$ for $i = 1,2,3$?

2. The host, who knows where the prize is, opens one of the 2 doors not chosen by the contestant and behind which the prize is not located. Let $A_2$ be the event "the host opens door 2" and $A_3$ the event "the host opens door 3". Calculate the probabilities

$$P(A_2 \mid D_1), \; P(A_2 \mid D_2), \; P(A_2 \mid D_3), \; P(A_3 \mid D_1), \; P(A_3 \mid D_2), \; P(A_3 \mid D_3).$$

3. Deduce $P(D_1 \mid A_2)$ and $P(D_3 \mid A_2)$. Similarly, what are $P(D_1 \mid A_3)$ and $P(D_2 \mid A_3)$?