Before each show, Monty secretly flips a coin with probability p of Heads. If the coin
lands Heads, Monty resolves to open a goat door (with equal probabilities if there is
a choice). Otherwise, Monty resolves to open a random unopened door, with equal probabilities. The contestant knows p but does not know the outcome of the coin flip.
When the show starts, the contestant chooses a door. Monty (who knows where the car is) then opens a door. If the car is revealed, the game is over; if a goat is revealed, the contestant is offered the option of switching. Now suppose it turns out that the contestant opens door 1 and then Monty opens door 2, revealing a goat. What is the contestant’s probability of success if he or she switches to door 3?
This is from "Introduction to Probability" By Joseph K. Blitzstein
The solution that I came up with is the following:
Ci – event that car is behind door i
H – event that coin landed head
Xi – event that participant picks door i initially
Oi – event that Monty opens door i
Given this we have:
$P(C3|X1, O2) = \frac{P(O2|C3, X1)P(C3|X1)}{P(O2|C3, X1)}$
And:
$P(O2|C3, X1) = P(O2|C3, X1, H)P(H|C3, X1) + P(O2|C3, X1, Hc)P(Hc|C3, X1)$
I'd appreciate some thoughts on this.
Best Answer
The coin flip, the door that the contestant picks, and the placement of the car will be mutually independent .
However, the door that Monty picked depended on their result. When the contestant and car indicate different doors, then if the coin landed heads, Monty certainly revealed the only goat he could, but otherwise he had unbiasedly choosen a door from those the contestant did not choose reguardless of the car placement.
$$\begin{split}\mathsf P(O_2\mid X_1,C_3) &= \mathsf P(H)~\mathsf P(O_2\mid H,X_1,C_3) + \mathsf P(H^\complement)~\mathsf P(O_2\mid H^\complement,X_1)\end{split}$$
Of course, if both the contestant and car choose the same door, then Monty's choiseof goats was unbiased whatever the coin may have said.
$$\begin{split}\mathsf P(C_3\mid X_1,O_2) &= \dfrac{\mathsf P(C_3)~\mathsf P(O_2\mid X_1,C_3)}{\mathsf P(C_1)~\mathsf P(O_2\mid X_1,C_1)+\mathsf P(C_3)~\mathsf P(O_2\mid X_1,C_3)}\end{split}$$