Brainteasers and unconditional love

The other day while browsing through the archives of Dean’s
World
I came across a fascinating little nugget of a post called The
Brainteaser That Changed My World
. I love brain teasers,
and this one was especially intriguing:

You find yourself on a game show called "Let’s
Make A Deal." The
game is very simple. There are three doors: door #1, door #2, and door #3.
Behind
one door is a million dollars. The other two doors contain worthless joke
prizes. All you have to do is pick which door you want to open, and you get
whatever
is behind it. But you only get to open one door. By simple math, then, you
obviously have a 1 in 3 chance of picking the correct door and becoming an
instant millionaire.

You
pick a door. As soon as you tell Monty (the gameshow host) what door you
want to open, he stops and says, "Okay, you’ve made your choice. Now, I’m
going to do what we always do here on this game. I’m going to open one of the
other two doors for you that I know has a booby prize." And he does so.
Then he asks, "Okay, now, would you like to stay with your original
guess, or would you like to switch to the other door that’s still closed?
You only
get one shot, so do you want to stay with your original choice, or switch?"

Here’s
the question: is there any compelling reason to switch doors?

Of course not,
I thought. Why would you switch doors? They both have a one in three chance
of being correct, or maybe even a one in two chance, but neither
has a clear advantage over the other.

But then I had an epiphany and thought, “Yes,
by all means switch doors.” Given
that you initially choose one door out of three, the door you choose clearly
has a one in three chance of being correct. The removed door has a zero in
three chance of being correct, which means the third door (the only other door
remaining)
must have a two in three chance of being correct. Nothing could be more obvious,
right?

I explained the problem to a few coworkers, and they all thought I was
insane for insisting that it’s best to switch answers. Dance with the
one that brung ya, they said, especially when there’s no reason to switch
partners, which there’s definitely not. When faced with my irrefutable
logic, they said I was arguing semantics, which doesn’t apply to mathematical
problems.

I don’t remember the last time so many people told me I was
wrong.

So
I came home and shared the story with my wife. She too came to the conclusion
that switching has no advantage over sticking with your first answer, but nevertheless
agreed to help me do a practical test. Before we could perform an experiment,
I was able to pick the brain of my father-in-law, a computer programmer who
has a degree in math. I thought surely he would agree with my conclusion. But
he
said the same thing everyone else did: “I see what you’re saying,
but I still think you’re wrong.” I should note, however, that my
mother-in-law agreed with me. I love her.

Anyway, Mrs. Happy and I set up the
experiment. I took three playing cards (taking the place of the three doors,
which we don’t have), one of which was the
ace of spades (taking the place of the million dollars, which we also don’t
have), and placed them face down. She chose one. I removed one of the other
two that was definitely not the ace of spades. She then changed her choice,
turned
over the remaining card, and marked down whether her first choice or second
choice was correct. We did that 100 times. The first choice turned up the ace
35 times,
and the second choice turned up the ace 65 times. Basically, the first choice
has a one in three chance of being correct while switching will win you the
money two out of every three times. So basically, I was right and everyone
but my mother-in-law
was wrong.

Here’s my point: My wife supported me and didn’t try
to make me feel stupid even when she thought I was dead wrong. She always left
open the
possibility
that I had come to the right conclusion and even took time out of her busy
day to help me perform a tedious test of an inconsequential problem, because
it was
important to me. That, my friends, is a good wife.

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