You are shown three boxes, and told that one of them contains a prize. You are then asked to pick one box, and if that box is the one with the prize, you will win it. After picking a box, you are shown that one of the other two boxes is empty, and offered a chance to change your selection.
Should you do this? Would changing your choice to the other remaining box affect your odds of winning? Why or why not?
If the person opening one of the unpicked boxes is
guaranteed to open an empty box, then we should switch. If the person randomly picks
from the remaining boxes and it happens to be empty, then it doesn’t matter
what we do. If this seems confusing, no worries; you have a lot of good company
in the math world.
The trick is to realize that there are two pieces of
information being given; (1) how the person is determining which box to open
and (2) the results of that opening. Let’s attack this via dissecting our
sample space into equiprobable events. Let’s number the boxes, starting with
the one we first chose. There are three, equally likely (1/3) ways the prize
can be situated:
1P, 2E, 3E (1/3)  1E, 2P, 3E (1/3)  1E, 2E, 3P
(1/3)
where E=Empty and P=Prize.
Assuming (1) the person is picking a box at random, these
events each split into 2 equally likely events:
1P, 2EX, 3E (1/6)  1P, 2E, 3EX (1/6)
1E, 2PX, 3E (1/6)  1E, 2P, 3EX (1/6)
1E, 2EX, 3P (1/6)  1E, 2E, 3PX (1/6)
where the X indicates the box the person opens.
Knowing that (2) the person did not reveal the prize
eliminates two of these events:
1P, 2EX, 3E (1/4)  1P, 2E, 3EX (1/4)
1E, 2P, 3EX (1/4)
1E, 2EX, 3P (1/4)
We see that there is a 1/2 chance of getting the prize,
which means switching is of no use.
Now assume (1) that you know the other person would not
reveal the prize box. We still have three initial possibilities:
1P, 2E, 3E (1/3)  1E, 2P, 3E (1/3)  1E, 2E, 3P
(1/3)
But before he even reveals a box, we know there is only one
possibility for each of the last two cases:
1P, 2EX, 3E (1/6)  1P, 2E, 3EX (1/6)
1E, 2P, 3EX (1/3)
1E, 2EX, 3P (1/3)
In this case, seeing the revealed box is empty (2) adds no
new information. Thus, there is a 2/3 chance of getting the prize if you
switch.
On the Monty Hall Show often a contestant was asked to pick
one of three doors, a goat (or some other lesser prize) was revealed behind one
of the remaining doors, and then the contestant was asked whether or not they wanted
to switch choices. It is assumed that for television purposes, the producers
would deliberately not open a door that hid the grand prize, so in this case
the contestant should switch. And actually, since there is no disadvantage to
switching in either case, contestants should switch regardless of their
certainty about how the producers are selecting the doors.

Posted by owl
on 20050817 14:36:31 