Game Development Reference
In-Depth Information
On the other hand, if we had picked a goat door to begin with, the host is now
in control of two doors… one that hides the car and one that hides the other goat.
The host is not going to reveal the car to us. Therefore, he
has to
open the door that
hides the other grass-muncher. Logically, that means the door that he
didn't
open
has the car.
At this point, our question may be, “But how do we know whether we selected
correctly to begin with?� Well, we don't. However, the solution is for us to use some
of the same simple probability math that we were willing to use before. Our error
was not doing this probability math sooner.
Rethinking the Probabilities
Originally, there were three doors from which to select. That means a random
choice (which was, in effect, what we were making) gives us a one-third chance of
selecting the car and a two-thirds chance of selecting a goat. Again, for emphasis, we
had a two-thirds chance of being wrong and only a one in three chance of being
correct in the first place. When brought forward to our second decision, the same
chances hold true. Regardless of what the host does in the interim, we still only have
a one in three chance of currently claiming the correct door. We have a two in three
chance of being the proud, temporary custodian of a domestic barnyard animal.
This is to our advantage, however. As we mentioned above, if we selected incor-
rectly (two-thirds chance), we forced the hand of the host. Of his two remaining
doors, he
must
take the other goat off the table and leave the car on it (which is a fun
metaphor, isn't it?). That means that if we originally selected one of the wrong
doors, we now know that the
other
remaining door is the car. It has to be.
So, to sum up:
We had a one-third chance of originally being right—we should stay with our
door in order to win.
We had a two-thirds chance of originally being wrong—we should switch
doors in order to win.
If we rephrase slightly to see what happens if we
always
switch, we arrive at:
With a one-third chance of originally being right—switching doors would
always
lose.
With a two-thirds chance of originally being wrong—switching doors would
always
win.
