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Sarah Bennett
Professor Winkler
Math 007
11 April 2013
Probability and Star-Crossed Lovers
There are few writing assignments that I distinctly remember, as the number of essays I
have had to complete is vast. An assignment about a forbidden romance with a tragic ending,
however, tends to attract the attention of most adolescent girls. The first time I heard this prompt
was during my seventh grade English class. The goal was to create an ending for a legend where
a princess’s lover faces a life or death decision. I was suddenly reminded of this past essay while
reading Martin Gardner’s The Colossal Book of Short Puzzles and Problems. Gardner created a
problem based upon this legend. He intertwines the suspense of the story with mathematical
principles of probability to create an intriguing puzzle. One must utilize wit and logic in order to
determine the fate of these star-crossed lovers and ultimately solve Gardner’s puzzle.
The legend begins in a kingdom under the reign of a cruel king. The king believed in the
power of fate, and the extreme nature of the kingdom’s judicial system reflected his belief. A
person charged with a crime was placed in a gladiator’s arena facing two closed doors at the
opposite side of the ring. Behind one door lies a hungry tiger; the other door conceals a fair
maiden. The demented king believed that if the person were truly guilty, the perpetrator would
choose to open the door with the tiger and then be immediately devoured. If, on the other hand,
the citizen had been wrongly accused, fate would guide him to choose the door hiding the fair
maiden. The citizen’s apparent innocence would be rewarded with an immediate wedding.
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The king’s cruelty extended to the treatment of his beautiful daughter. He had forbidden
her to court any of her suitors, hoping to deny her true love and keep her in the castle with him
forever. Despite the king’s mandate, his daughter fell in love. The love between the princess and
her suitor was pure and true, so the pair attempted to flee the kingdom to be together. In the
midst of their escape, the king intercepted the lovers. The cruel king was enraged and brought the
young man to trial immediately. The princess’s lover then faced a dire decision, but luckily the
clever princess had discovered which door held the tiger and which concealed the young maiden.
Distraught, she knew her lover would either be devoured or be wed to another woman. The
young man looked to the princess before choosing a door. The princess began to cry as she lifted
her delicate hand and discreetly pointed to the right. The legend then ends abruptly.
Martin Gardner expands upon the legend in his puzzle by constructing a more elaborate
justice system. He proposes that the person being tried must make three consecutive choices of a
fair maiden in order to be proclaimed innocent. If the citizen ever chooses a tiger, then the trial
ends immediately as the person is eaten and presumed guilty. Gardner’s system is comprised of
three pairs of doors. One pair hides two tigers, one pair hides a maiden and a tiger, and the last
pair hides two completely identical maidens. The first choice is to pick a single door from one of
the pairs. If a maiden is behind the door, the trial continues using only the chosen pair of doors;
the other two pairs are ignored. To begin the next choice, the citizen steps back and the door is
shut once again. The two things in the pair are once again randomly placed behind the two doors.
The citizen must once again select a door. If this choice reveals another maiden, the doors are
closed and the pair is randomly assigned to a door for a final time. If the citizen then selects a
door that conceals a maiden for a third time, then a wedding occurs and the accused individual is
proclaimed innocent.
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After explaining his elaborate addition to the legend, Gardner states his belief that the
lovely princess was not a jealous woman. He therefore presumes that she guided her lover in his
first choice so that he was able to choose a door with a maiden behind it. Now the young man
has a single pair of doors from which he could make selections, but with two decisions
remaining, the young man is still in danger. Luckily, upon his second choice, the young man
managed to select another maiden. Gardner then proposes a simple question to the reader. What
are the young lover’s chances of survival if his first two choices reveal a maiden?
When I first began to solve this puzzle, I racked my memory for knowledge on
probability. If the first door was a maiden, then he either chose the pair of doors that concealed
the two identical maidens or the one that concealed a maiden and a tiger. The pair of doors that
concealed two tigers was immediately eliminated as a possibility by his first choice. I then
remembered that he must choose a maiden three consecutive times in order to survive his trial. I
noticed that three of the four potential choices were maidens. The probability of randomly
selecting a maiden would be 75%. I then thought I had found the probability of the young man
surviving and therefore solved the puzzle.
This reasoning was not completely logical. I had incorrectly allowed the probability to
include four equally likely outcomes: choosing one of three different maidens or a tiger. In
reality, the young man had already selected a pair of doors and therefore eliminated four of the
six original doors. Therefore it is impossible to have four potential outcomes as a pair of doors
conceals only two things. To solve this problem, one needs to simply list the sample space for
combinations that ensure three choices and uses only one pair of doors (this means the first two
choices cannot consist of a tiger). If the young man chose the pair of doors hiding two identical
maidens, the possibilities are as follows:
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Lady 1—Lady 1 – Lady 1
Lady 1 – Lady 1 – Lady 2
Lady 1 – Lady 2 – Lady 1
Lady 1 – Lady 2 – Lady 2
Lady 2 – Lady 1 – Lady 1
Lady 2 – Lady 1 – Lady 2
Lady 2 – Lady 2 – Lady 1
Lady 2 – Lady 2 – Lady 2
If the young man had selected the pair of both a maiden and a tiger, then the only combinations
where the first two choices resulted in a maiden are:
Lady 3 – Lady 3 – Lady 3
Lady 3 – Lady 3 – Tiger
This sample space contains ten possibilities. Of these ten possibilities, nine result in the young
man surviving, 90%, a probability greater than my initial idea of 75%.
I was interested in this puzzle because of its elegant simplicity. The solution is simply the
sample space of combinations that lead to three choices. Furthermore, the human side of me
wants the young man to survive his endeavor. I enjoyed the prospect of a mathematical challenge
and a high probability of a happy ending.