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The solution given by the Car Talk boys has the right idea but they struggle a bit with the mathematics. Also the problem is not well defined.  However this is a well-known problem.  It is problem 20 in Mosteller's famous book  "Fifty Challenging Probability Problems". Here is his description of the duel as applied to the Car Talk problem.
==What are the odds of that?==


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 the Car Talk boys suggest that A should make is first shot to the sky and otherwise shoot at the most skilled of the duelers still alive.  Of course suggesting that A shoot at the sky is counterintuitive this is what makes the problem interesting. Neither Mosteller or the Car Talk when defining the game say that shooting at the sky is an option but Mosteller 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 duelers wins.  Under his strategy the possible paths the duel are shown in the tree diagram below.
113,527,276,681,000,000 to 1
Those were the odds local golfer Jacqueline Gagne could
make seven holes in one in 14 weeks.
The Desert Sun
Palm Springs
Larry Gohannan
April 28, 2007


The nodes give the remaining duelers. The bar indicates the dueler who shoots next. The branches have the probabilities for the possible outcomes of a shot.  All but one of these probabilities is obvious from skill of the dueler. The one that is not obvious when A has only B to shoot at.  In this case we have a two-person duel and we assume that A shoots first and the duelers alternate shots until one is killed.
Lee Sechrest suggested this would be a good example to discuss how we should react to a claim that an event with an  unbelievable small probabillity occurred.


Let p be the probability that A wins and q the probability that B wins this two person duel.  Then A wins if he hits B with his first shot (probability 1/3).  If A and B both miss their first shots  (probability 2/3*1/3 = 2/9 then A still has probability p of winning.  Thus p = 1/3 + 2/9p.  Solving for p we have p = 3/7 and so q = 4/7.
Jacqueling Gagne is a 47 year old who retired from a computer progrem job and moved to


<center>http://www.dartmouth.edu/~chance/forwiki/duel.jpg</center>
<blockquote>Before she recorded her eighth this week, Professor Mike McJilton of the College of the Desert Math and Science Department calculated the odds of making seven aces in approximately 65 rounds (five rounds a week for 13 weeks) as Gagne did is just under 114 million billion to 1. As a decimal number, the chance of making the seven aces is 0.0000000000000000088
(8.8*10^-18).</blockquote>


Now we can determine from the tree each player’s chance of winning the duel. Note that there is only one branch resulting in C winning and the probability that this branch occurs is 1/3*2/3 = .2/9 = 222. There is also only one branch that will result in B winning and this branch has probability 2/3*4/7 = 8/21 = .381. Thus the probability at A wins is 1-8/21 - 2/9 = 25/63 = .397.  
Another article said that McLilton used the poisson distribution presumeably as an opproximation to the binomial distribution. For this he needed an estimate for the probability that a reasonable golfer makes a hole in one round.  This probability has been estimated to be 1/5000. However using  Mathematca it is easy to compute the probability using the binomial distribution with p = 1/5000 and n  = 65. Doing this find the probability getting seven success to two decimal places  in 65 rounds is 8.8x10^-18 in agreement with Mclilton’s result.


From this we see that A has the highest probability of winning and C the lowest probability. Thus we have survival of the unfitist.
The next mention of calculating the odds was in the ESPN sports news May 24, 2007. Here we read
 
<blockquote> We asked one of the world's foremost mathematicians, Joseph B. Keller, professor emeritus of mathematics and mechanical engineering at Stanford, to compute the odds of making 10 aces in 80 rounds (using a previous finding by Golf Digest that the odds of an average player making an ace in any given round are 1 in 5,000). Keller's answer: "Roughly two chances out of 10 followed by 24 zeros. (2/10^25) This is the same chance as picking one particular molecule out of all the molecules in 50 gallons of air."
 
Mathematica gives 1.7/10^25 in agreement with Keller’s estimate.
 
By June 3 Jacqueline  had made 14 holes in one in the past 5 months. Again using the Binomial model, the probability of this event is  3x10^-35
 
Of course people were suspicious of a fraud but they were able to find witnesses to all of her hole in ones.  Her authenticity gained further support when a TV station sent a crew to interview her and she hit a hole in one while they were there.
 
An interesting discussion of what to make of all this is found in an article by  CARL BIALIK  JUST HOW AMAZING IS JACQUELINE GAGNE -- AND HER 10 GOLF ACES? Section B, Column 1,  May 18, 2007
 
The most interesting discussion of this event is

Revision as of 19:57, 27 June 2007

What are the odds of that?

113,527,276,681,000,000 to 1
Those were the odds local golfer Jacqueline Gagne could make seven holes in one in 14 weeks. The Desert Sun Palm Springs Larry Gohannan April 28, 2007

Lee Sechrest suggested this would be a good example to discuss how we should react to a claim that an event with an unbelievable small probabillity occurred.

Jacqueling Gagne is a 47 year old who retired from a computer progrem job and moved to

Before she recorded her eighth this week, Professor Mike McJilton of the College of the Desert Math and Science Department calculated the odds of making seven aces in approximately 65 rounds (five rounds a week for 13 weeks) as Gagne did is just under 114 million billion to 1. As a decimal number, the chance of making the seven aces is 0.0000000000000000088 (8.8*10^-18).

Another article said that McLilton used the poisson distribution presumeably as an opproximation to the binomial distribution. For this he needed an estimate for the probability that a reasonable golfer makes a hole in one round. This probability has been estimated to be 1/5000. However using Mathematca it is easy to compute the probability using the binomial distribution with p = 1/5000 and n = 65. Doing this find the probability getting seven success to two decimal places in 65 rounds is 8.8x10^-18 in agreement with Mclilton’s result.

The next mention of calculating the odds was in the ESPN sports news May 24, 2007. Here we read

We asked one of the world's foremost mathematicians, Joseph B. Keller, professor emeritus of mathematics and mechanical engineering at Stanford, to compute the odds of making 10 aces in 80 rounds (using a previous finding by Golf Digest that the odds of an average player making an ace in any given round are 1 in 5,000). Keller's answer: "Roughly two chances out of 10 followed by 24 zeros. (2/10^25) This is the same chance as picking one particular molecule out of all the molecules in 50 gallons of air."

Mathematica gives 1.7/10^25 in agreement with Keller’s estimate.

By June 3 Jacqueline had made 14 holes in one in the past 5 months. Again using the Binomial model, the probability of this event is 3x10^-35

Of course people were suspicious of a fraud but they were able to find witnesses to all of her hole in ones. Her authenticity gained further support when a TV station sent a crew to interview her and she hit a hole in one while they were there.

An interesting discussion of what to make of all this is found in an article by CARL BIALIK JUST HOW AMAZING IS JACQUELINE GAGNE -- AND HER 10 GOLF ACES? Section B, Column 1, May 18, 2007

The most interesting discussion of this event is

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