Example 2: The Monte-Hall Game Show

You are about to choose your winning in a game show. There are three doors behind one of which is a red Porsche and other two, goats. You will get whatever is behind the door you choose. You pick a door, say A. At this point the game show host opens one of the other two doors, which he knows to contain a goat, for example B and asks if you would now like to revise your choice to C. The question is: Should you? (Assuming you want the car and not the goat.)

Let PX be the event that the Porsche is behind door X and Let GX be the event that a goat is behind door X.

$$\begin{eqnarray*}P(PA\vert GB) &=& \frac{P(PA) \cdot P(GB\vert PA)}{\sum\limits_... ...(PX) \cdot P(GB\vert PX)}\\ &=& 1/(1/2 + 0 + 1)\\ &=& 2/3\\ \end{eqnarray*}$$

For homework, do a Monte-Carlo simulation of the game show showing that after a large number of trials, P(PC|GB) does indeed approach 2/3.