CS 2800: Discrete Math
Oct. 17, 2011
Lecture
Lecturer: John Hopcroft
Scribe: Dan & June
Review
1
Monte Hall Problem
The Monte Hall Problem is an interesting excercise in conditional probability. In the scenario, you are on
a game show competing for a grand prize. The game consists of you choosing among three doors and you
are awarded whatever prize hides behind the door you choose. You know that one of the doors is hiding a
sports car and the other two are hiding goats. Obviously, it is your objective to choose the door hiding the
car. However, you do not know which door is hiding what prize.
When the game begins you select a door (let’s say door A), but it is not opened yet. Because there are
two goats, at least one of the two doors you did not choose must be hiding a goat. So, as an added twist,
the host of this show  Monte Hall  will open this door (let’s say door B) and reveal a goat. He then poses
this question: “ Do you wish to change your selection to the other remaining door (door C)? ”
What do you do?
Many people will first assume that switching doors will neither help or hurt your
chances. Their reasoning is that when you initially choose, each door has a 1
/
3 chance of hiding a car and
Monte revealing one door doesn’t change this uniform probability.
This turns out to be
false
.
In reality,
you double your chances of winning the car if you choose to switch. For a mathematical explanation, you’ll
need to turn to conditional probability and Bayes Theorem (covered in the next lecture). However, we can
develop some intuition to convince you of this startling effect.
Let’s consider the only two cases. Case 1 is when you initially choose a door with a goat behind it. Then,
Monte will reveal the only remaining goat and, clearly, you should switch to the final door (which must be
hiding the car). In Case 2, you initially choose the door with the car behind it. Then, Monte will reveal
either of two remaining goats and ask you if you wish to switch. Obviously, you will lose the car if you switch
and it is better to keep your first choice.
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 '07
 SELMAN
 Conditional Probability, Probability, Playing card, MONTE, Monte Hall

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