Does someone know a solution to the following generalization of the Monty Hall Problem:
The Problem: Assume you are on Let's Make a Deal and are presented with the regular dilemma of the Monty Hall Problem. Unlike normally, however, the car isn't placed randomly behind a door, but according to a certain probability. So the probability that the game show host puts the car behind door $1$ is $A_1$, behind door $2$ is $A_2$ and behind door $3$ is $A_3$. (With $A_1+A_2+A_3=1$, of course)
Additionally, you have a preference for choosing some of the doors over others, meaning that the probability of you choosing a random one of the three doors on your first try is not necessarily $1/3$. So the probability of you choosing door $1$ is some $B_1$, the probability for choosing door $2$ is $B_2$ and the door $3$ has probability $B_3$ of being chosen by you. (Once again with $B_1+B_2+B_3=1$)
Now what is the probability of you winning the car?
As the normal Monty Hall Problem is already tough enough for me, I thought I might ask you all for help. If my notations are bad please do not hesitate to change them. Nonetheless, my progress so far:
My Progress: Obviously the probability of choosing the $1st$ door and the car being behind it is $A_1\cdot B_1$ if you do not switch. The probability of you choosing the $2$nd door and finding the car behind it if you do not switch is $A_2\cdot B_2$ and the accordingly the probability of finding the car behind door $3$ without switching is $A_3 \cdot B_3$.
And this is right about where my head starts spinning, because I don't know where to go from here on