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Pretend you are on a game show with Monty Halland he
offers you the following scenario as described in a 2003 article from the Journal of Experimental Psychology.
**You
face three doors and behind one door is a car, while the other two hide goats. Your
goal is to pick the door that hides the car. Here are the rules. First, the car
and the goats were placed randomly behind the doors. Second, after you choose a
door, the door remains closed for now. Third, Monty knows what is behind each door.
Fourth, he has to open one of the two remaining doors. Fifth, the door he opens
must have a goat behind it. Sixth, if both remaining doors have goats behind
them, he chooses one randomly. After
Monty opens a door with a goat, he will ask you to decide whether you want to stay
with your first choice or switch to the last remaining door. Pretend you chose
Door 1 and Monty opens Door 3 containing a goat. With only Doors 1 and 2
remaining-one of which contains a car-he asks you, "Do you want to switch to
Door 2?" From
a probability standpoint, are you more likely to win the car by staying with
your original choice of Door 1, switching to Door 2, or does it make any
difference at all if you stay or switch? Before reading further, think of your
answer then return to the next paragraph.
As you
contemplated your answer, you may have reasoned that since one of the two
remaining doors contains the car, you have a 50/50 chance of winning, so there
is no need to switch. That may sound reasonable, but it is not correct. Presented
with this three-door scenario, you should always switch, in fact, by switching,
you have a 2/3 probability of picking the car.
Here's the
explanation, according to Michael Shermer writing in the February 2009 issue of
Scientific American.
**At
the beginning of the game you have a 1/3 chance of picking the car and a 2/3
chance of picking a goat. Switching doors is bad only if you initially chose
the car, which happens only 1/3 of the time. Switching doors is good if you
initially chose a goat, which happens 2/3 of the time. Thus, the probability of
winning by switching is 2/3, or double the odds of not switching.
Over countless
studies using this "Monty Hall" problem, the vast majority of participants
think that staying and switching are equally good alternatives. So, if you are
in that camp, you have lots of company.
For investors,
the fact that the majority of people who take the "Monty Hall Challenge" get it
wrong suggests that there may be times when the majority of investors are
"wrong," too. At crucial turning points in the stock market, when there is
evidence to support two opposite directions for the major averages, the
majority of investors may "misread" the data (as in the Monty Hall Challenge) and
draw a conclusion that subsequently turns out to be incorrect. While we will
not always be "smarter" than the crowd, we do realize that, like the Monty Hall
problem, the crowd is not always right. And because of our open mind, our
willingness to think differently, we are constantly scanning for opportunities or
turning points that may be overlooked by the crowd.
Sidebar: Are you still shaking your head
about the answer to the "Monty Hall" problem? Here's another way to look at it
from mathforum.org.
What if there were 1,000 doors? You would
initially have a 1/1,000 chance of picking the correct door. If Monty opens 998
doors, all of them with goats behind them, the door that you chose first will
still have a 1/1,000 chance of being the one that conceals the car, but the
other remaining door will have a 999/1,000 probability of being the door that
is concealing the car. Here switching sounds like a pretty good idea.
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