A scratch game has 16 covered squares, below which are, in some order, symbols representing 9 players (a pitcher, a catcher, and so on), 3 strikes (strike 1, strike 2, and strike 3), 1 hit, 1 home run, 1 bat, and 1 ball.

You start scratching squares randomly, and if you manage to get the hit and the home run you win (never mind other symbols) unless you already got the three strikes, because then you're out!

What are the odds of winning at this game?

Computer solutions welcome!

First, I'd like to propose the removal of all purposeless symbols from our considerations.  It may not be immediately obviously to everyone, but they do not at all affect the calculations.

Let's just say that I scratch all the squares, rather than stopping when I win or lose.  I can take any of the 5! possible permu
tations, each of which is equally likely, and determine whether I win or lose first by looking at the last square I scratch.  If the last square is a strike, I've won.  If the last square is a hit or home run, then I've lost.  Exactly 3/5 of the 5! permutations will cause me to win.  Therefore, I have exactly 60% chance to win.

Since bold lettering always stands out, I've also written a java program to estimate the probability.  I don't yet know how to calculate standard deviations though. :(

// ScratchBaseball.java
// Estimates the probability of winning a game of scratch baseball
// Tristan 10/18/05

// java packages 
import javax.swing.JOptionPane;
import javax.swing.JTextArea;

public class ScratchBaseball
   {
   public static void main( String args[] )
      {
      String repeat = ""; // User's choice whether to repeat
      JTextArea output = new JTextArea();
      
      final int numberTrials = 100000; 
      // number of trials to perform for each repeat of the program
      int trial = 0; // trial counter
      int trialTotal = 0; // total number of trials performed
      int squares = 16; // total number of unscratched squares
      int homerun = 0; // number of winning squares scratched
      int strike = 0; // number of strikes scratched
      int winCtr = 0; // winning trials
      int loseCtr = 0; // losing trials
      int guess = 0; // square to be scratched
      
      do
         {
         trial = 0;
         
         while( trial < numberTrials )
            {
            squares = 16;
            homerun = 0;
            strike = 0;
                       
            while( homerun < 2 && strike < 3 )
               {
               guess = 1 + (int) ( Math.random() * squares-- );
               
               if( guess <= 2 - homerun )
                  homerun++;
               else if( guess <= 5 - homerun - strike )
                  strike++;
               }
            
            if( homerun == 2 )
               winCtr++;
            else
               loseCtr++;
            
            trial++;
            }
            
         trialTotal += numberTrials;
         
         output.setText( "Trials:\t" + trialTotal + 
            "\nWins:\t" + winCtr + "\nLosses:\t" + loseCtr + 
            "\nWin probability:\t" + 
            ( (double) winCtr / (double) trialTotal ) );
         
         JOptionPane.showMessageDialog( null, output, "Results",
            JOptionPane.PLAIN_MESSAGE );
         
         repeat = JOptionPane.showInputDialog( 
            "Type 0 if you would like to repeat the program" );
         
         } while( repeat.equals( "0" ) );
         
      System.exit( 0 );
      
      } // end main method
   } // end public class

Results
Trials:    100000
Wins:    60069
Losses:    39931
Win probability:    0.60069

Posted by Tristan on 2005-10-18 20:03:51