monty hall
Post on 30-Jun-2015
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THE MONTY HALL
WELCOME My name is Ercan Cem
I will try to present
the most famous* puzzle of all times.
the most famous* puzzle of all times.
*arguable
It is known as The Monty Hall
POPULAR BECAUSE
COUNTER INTUITIVE
THE PUZZLE
You are in a TV show.
There are three doors.
1 2 3
Behind one of them there's a prize.
It is yours if you guess correctly.
The format of the show is...
First, you make a guess.
1 2 3
The host opens an empty door from the other two.
1 2 3
The host knows where the prize is.
He offers you a new option:
You can stick with your initial choice.
OR
Select the other (unopened) door.
QUESTION
WHAT IS THE RATIONAL
STRATEGY?
IS IT...
Stick?
1 2 3
OR...
Select the other?
1 2 3
OR...
Is this just a random guess?
1 2 3
The usual* answer is that the final choice is just a random guess.
The usual* answer is that the final choice is just a random guess.
*underestimate
People tend to think that
at first, each door has the same one in three chance.
(which is true)
So, the chance that our guess is correct
is one in three.
(which is true)
Once one of the doors is opened
1 2 3
the unopened two doors
1 2 3
share the chance of the opened door.
THEREFORE
The final decision is a random guess.
1 2 3
WRONG!
The answer is:
If you want to increase your chance,
you must switch to the other door.
1 2 3
(we exclude the cases)
(that you are superstitious about your inital guess)
OR
You are a clairvoyant.
HERE IS THE EXPLANATION
(without getting too technical)
FIRST TRY
When you made your initial guess,
1 2 3
your chance of being correct was
one in three.
ALSO,
As a fact, we know that
at least one of the remaining
doors is empty.
AGREE?
GOOD.
FURTHERMORE,
The host will always have a choice to open
an empty door.
SO,
when he opens an empty door,
1 2 3
he does not provide an extra information.
He does not provide anything new.
We already know that at least one
other door is empty.
The chance that our initial guess is correct
is one in three.
Nothing changed since then.
After the door is opened,
that chance is still one in three.
HOWEVER
When two doors are left,
it cannot be the case that each has a chance
of one in three.
The TOTAL chance must add up to 1.
THEREFORE
The chance that the prize is behind the other
door is two in three.
UNLESS
You are very supertitious,
OR
CLAIRVOYANT
You must switch.
Not convinced?
SECOND TRY
Imagine that just before the host opens one of the remaining doors,
the aliens kidnap him.
Right at this moment, we can reason as follows:
The host would either open Door- 2, or Door -3.
Say he opened Door -2
1 2 3
Then, only Door -1 and Door-3 would be left.
1 2 3
The chance that the prize is behind Door-1
would be 50%.
Instead, say he opened Door -3
1 2 3
Then, only Door -1 and Door-2 would be left.
1 2 3
The chance that the prize is behind Door-1
would be 50%.
Hmmm!
Isn't that reasoning a bit...
NAIVE?
BECAUSE
It suggests that
the chance that our initial guess is correct was 50% in the first place.
We KNOW that that is WRONG!
It is one in three.
It cannot increase all of a sudden.
THEREFORE
The chance that the prize is behind the other
door is two in three.
Not convinced?
LAST TRY
LAST TRY (You’d better be convinced this time.)
Imagine a deck of cards.
(52 cards that is.)
You pick one.
If it is the ace of spades,
you win.
You picked one.
51 cards are left.
I KNOW which card is the ace of spades.
Among the other 51 cards,
at least 50 of them is NOT the ace of spades.
I turn 50 cards upside down.
Right now, there are two unopened cards on the table...
yours, and the 52nd card.
Do you REALLY
think that
the chance that your pick is the ace of spades is 50%?
TIME TO WRAP UP
In the last example, the essence of the
puzzle is much clearer.
That is because we were dealing with larger quantities.
Things look fuzzier when dealing with only three objects.
SIDE NOTE
In the original problem, suppose that the host opens a random door.
(Translation: It could be the case that the opened door is the one with the prize.)
If the opened door is empty,
then the chance that our initial guess is correct indeed rises to one in two.
HENCE
Everything depends on
whether the door was opened at random,
or with the knowledge that where the prize is.
Thanks for sparing your time.
(A simple Google search will get you to endless sources on Monty Hall. There’s even a book devoted on it.)
(If you enjoyed this puzzle, you can find more at my blog Mathzzle: puzzle.ercancem.com.)
UNTIL NEXT TIME
SEE YA!
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