Nice sketch Rob
I have just stuck some names on the results for disbelievers :-
//Â Â FILE: simuBarGame.ino
//Â AUTHOR: Rob Tillaart
// VERSION: 0.1
// PURPOSE: simulate ashtray game
//Â Â URL: http://forum.arduino.cc/index.php?topic=182297.0
//
// Released to the public domain
//
unsigned long win1 = 0;
unsigned long win2 = 0;
unsigned long games = 0;
float ratio = 0.0;
void setup()
{
 Serial.begin(115200);
 Serial.println("Start ashtray game");
}
void loop()
{
 games++;
 int pos = random(3); // position of coin
 int initialChoice = random(3); // initial choice
 // NOT CHANGING PLAYER
 // straightforward
 if (pos == initialChoice) win1++;
 // CHANGING PLAYER
 // select one that is empty to turn
 int turn = 0;
 for (int i = 0; i < 3; i++)
 {
  if (i == pos) continue;
  if (i == initialChoice) continue;
  turn = i;
 }
 // change choice
 int newchoice = 0;
 for (int i = 0; i < 3; i++)
 {
  if (i == turn) continue;
  if (i == initialChoice) continue;
  newchoice = i;
 }
 if (pos == newchoice) win2++;
 ratio = (1.0 * win1)/win2;
 // DISPLAY
  Serial.print("games played =");
 Serial.print(games);
 Serial.print("\t");
 Serial.print("won by keeping =");
 Serial.print(win1);
 Serial.print("\t");
 Serial.print("won by changing =");
 Serial.print(win2);
 Serial.print("\t");
 Serial.print("ratio =");
 Serial.println(ratio, 4);
}