Car Talk Shootout Puzzler

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Problem Description

Three men who are mutual enemies decide to settle things with a shootout. Al is a poor shot only hitting his target 1/3 of the time. Bob hits his target 2/3 of the time and Charlie hits what he aims at 100% of the time.  To even the odds a bit, Al is given first shot. Bob is next, if he's still alive. He's followed by Charlie, if he's still alive. They will continue shooting like this, in this order, until two of them are dead.

The question is: At whom should Al aim his first shot to maximize his chances of surviving?

Background & Techniques

This problem was borrowed from a recent Puzzler on the Car Talk radio program available here.  When math skills fail, programming skills can (sometimes) save the day.   That was the case here.  I decided to simulate lots of shootouts under various strategies and empirically find the best one.   The most obvious plan would be to shoot at the guy most likely to kill you.  The only other two plans I could think of were to shoot at the fellow least apt to kill you, and to choose your target randomly, as if you did not know  the accuracy statistics of each shooter.   It turns out that there is a fourth, the winning, strategy, as I found when I checked Car Talk's solution.   So I added that choice a well, but I won't  describe it here (in case you want to work on the problem on your own first).

The program allows you to select a strategy, then click a button to conduct 1,000,000 shootouts and display the results.    The shot-by-shot results  are displayed for the first 100 contests.  Each contest has each shooter in turn apply the  chosen strategy to select a target,  then record a "hit" based on that shooters probability of hitting his target.   (For example, for Al's accuracy score of  1/3, we generate a random  real number between 0 and 1 and record a hit if the number is less than 0.3333.  This technique is called a "Monte Carlo" simulation.)      

  Non-programmers are welcome to read on, but may want to skip to the bottom of this page to download executable version of the program.

Programmer's Notes:

The program defines a shooter record, TShooterRec, for each participant.  The record contains the constant information about the shooter (name and accuracy score), information about his current status within a game (whether alive or not), and  his total statistics for this set of shootouts (number of wins and number of hits).

And array of these records, Recs,  is initialized for each "SolveBtn" click.  A loop then runs the 1,000,000 shootouts.  "Alive" status is set to true for all participants before each shootout.  The "TakeAShot" procedure is a recursive routine which takes a shooter array index, Shooter,  and the Shotnbr  as parameters.   It performs these operations

bulletDetermine the target
bulletTake the shot and record whether or not target was hit if one is found.  A hit is determined by comparing a random real number between 0 and 1 to the  shooters accuracy rating.   "Less than" counts as a hit.  If hit, the shooter's hit count is incremented and the target's  "alive" status is changed from true to false.  
bulletIdentify next shooter, NextShooter,.   Since shooters take turns in "round robin" fashion,  we search up through the Recs array from the current shooter's index number looking for a living participant.  (Round robin" means that the first shooter follows last shooter in shooting order.)
bulletIf the global ShowDetail variable is true( in the first 100 games), we display the result s for this shot.
bulletFinally, if no next shooter was found, the game is over and we can update total statistics and display winner information if ShowDetail is true.  If a next shooter  was found, we call ourselves again passing NextShooter and ShotNbr+1 as parameters.

That's about it.  After the 1,000,000 shootouts, final stats are displayed.

Addendum February 21, 2012:  Version 2 posted today adds a second version of the Car Talk puzzle with different accuracy probabilities  for the three shooters.  This version is from a book Fifty Challenging Problems in Probability Frederick Mosteller, Dover Publications which I am currently working my way through.  I had previously posted several other problems from the book in our math section as Fifty Probability Problems.  For each of those I included an analytical as well as an experimental solution but I have yet to solve this one analytically.  I decided to add the experimental result here to provide some clues to the correct theoretical solution.  

Running/Exploring the Program 

bullet Download source
bullet Download  executable

Suggestions for Further Explorations

Number of shootouts and number of detailed results per trial could be user inputs, although  1,000,000  and 100 seem to be good enough numbers to verify the strategy chosen and to get consistent results taking only a few seconds  to run..  

 

Original Date: February 13, 2007

Modified: February 18, 2016

 

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