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The solution given by the Car Talk boys has the right idea but they struggle a bit with the  mathematics.  This is a well known problem and occurs in most math puzzles books.  It is problem 20 in Mosteller's famous book  "Fifty Challenging Probability Problems". We will present his proof addaped to the Car Talk problem.
The solution given by the Car Talk boys has the right idea but they struggle a bit with the  mathematics.  This is a well known problem and occurs in most math puzzles books.  It is problem 20 in Mosteller's famous book  "Fifty Challenging Probability Problems". We will present his proof addaped to the Car Talk problem.


A, B, and C are to fight a three-cornered pistol duel. All know that A's chance of hitting his target is 1/3, B's is 2/3, and C never misses. They are to fire at their choice of target in succession in the order A, B, C, cyclically (but a hit man loses further turns and is no longer shot at) until only one man is left unhit.  What should A’s strategy be.
A, B, and C are to fight a three-cornered pistol duel. All know that A's chance of hitting his target is 1/3, B's is 2/3, and C never misses. They are to fire at their choice of target in succession in the order A, B, C, cyclically (but a hit man loses further turns and is no longer shot at) until only one man is left unhit.  What should A’s strategy be?


Both Mosteller and Car Talk give arguments to suggest that for his first shot A should shoot into the air and in all other cases when a dueler has to choice between two dueler to shoot at he should choose the more skillful of the two.   
Both Mosteller and Car Talk give arguments that for his first shot A should shoot into the air and in all other cases when a dueler has to choose between two duelers to shoot at he should choose the more skillful of the two.  The fact that for his first shot  A should not shoot at anyone is considered counter-intuitive. Note that in his formulation of the problem Mosteller did not state that dueler could shoot into the ski.  He Writes
<blockquote>In discussing this with Thomas Lehrer, I raised the question whether that (shooting at the ski) was an honorable solution under the code duello.  Lehrer replied that the honor involved in three-cornered duels has never been established and so we are on safe ground to allow A a deliberate miss. </blockquote>


 
If we assume that the Duelers use the strategy suggested by Mosteller then we can compute the probability that each of the three survives.  Under his strategy the possible paths the duel could follow are shown in the tree diagram below:
<center><math>(1/3)(1/3) + (1/3)^2(2/3)(1/3) + (1/3)^3(2/3)^2(1/3) \cdots</math></center>
 
Each term corresponds to a sequence of misses by both B and A ending with a final hit by A.
 
Summing the geometric series we get
 
<center><math>(1/3)(1/3)[1 + (1/3)(2/3) + [(1/3)(2/3)]^2 + \cdots] </math></center>
 
<math>= \frac{(1/3)(1/3)}{(1-(1/3)(2/3)} =  \frac {1}{7} < \frac{1}{3}</math>
 
Thus hitting C and finishing off with B has less probability of winning for A than just misssing the first shot.  So A fires his first shot into the ground and then tries to hit C with is net shot.  C is out of luck.
 
Mosteller writes:  <blockquote>In discussing this with Thomas Lehrer, I raised the question whether that was an honorable solution under the code duello.  Lehrer replied that the honor involved in three-cornered duels has never been established and so we are on safe ground to allow A a deliberate miss. </blockquate>

Revision as of 18:20, 13 March 2007

The solution given by the Car Talk boys has the right idea but they struggle a bit with the mathematics. This is a well known problem and occurs in most math puzzles books. It is problem 20 in Mosteller's famous book "Fifty Challenging Probability Problems". We will present his proof addaped to the Car Talk problem.

A, B, and C are to fight a three-cornered pistol duel. All know that A's chance of hitting his target is 1/3, B's is 2/3, and C never misses. They are to fire at their choice of target in succession in the order A, B, C, cyclically (but a hit man loses further turns and is no longer shot at) until only one man is left unhit. What should A’s strategy be?

Both Mosteller and Car Talk give arguments that for his first shot A should shoot into the air and in all other cases when a dueler has to choose between two duelers to shoot at he should choose the more skillful of the two. The fact that for his first shot A should not shoot at anyone is considered counter-intuitive. Note that in his formulation of the problem Mosteller did not state that dueler could shoot into the ski. He Writes

In discussing this with Thomas Lehrer, I raised the question whether that (shooting at the ski) was an honorable solution under the code duello. Lehrer replied that the honor involved in three-cornered duels has never been established and so we are on safe ground to allow A a deliberate miss.

If we assume that the Duelers use the strategy suggested by Mosteller then we can compute the probability that each of the three survives. Under his strategy the possible paths the duel could follow are shown in the tree diagram below:

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