Smiles of a Summer Night

Smiles of a Summer night is been a favourite film of mine since I was at university.  I recommend everyone to watch it.  I just watched it again this morning and noticed something trivial.  When the soldier suggests to the lawyer to play Russian roulette - a version where each participant gets two goes each, and you spin the barrel to see who goes first - he mentions that the odds are '12 to 2' (based no doubt on the pre-Cardano assumption of two times six-to-one amounts to two in twelve).

Never trust a soldier.  Two scenarios.  One, you go first.  Two, your opponent goes first. 

 If you go first, the chance of immediate death is $\frac{1}{6}$.  But your chance of death on your second go depends on whether your opponent dies during his go and assumes you survived the first attempt.  There's a $\frac{5}{6}$ chance you make it through round one and then same again that he makes it through and forces you to fire again.  So your chances of death on the second pull of the trigger is $\frac{5}{6}\frac{5}{6}\frac{1}{6}$ or $\frac{25}{216}$.  Total chance of death is  $\frac{1}{6} + \frac{25}{216} = \frac{61}{216}$

If you go second then chances of your death on your first trigger pull is $\frac{5}{6}\frac{1}{6}$ or $\frac{5}{36}$.  Death on second trigger pull is $\frac{5}{6}\frac{5}{6}\frac{5}{6}\frac{1}{6}$ or $\frac{125}{1296}$ to give a going-second total of $\frac{5}{36} + \frac{125}{1296}$ or $\frac{180+125}{1296}$ or $\frac{305}{1296}$

If you spin the gun barrel to see who goes first then your total chance of death is $\frac{1}{2}\frac{61}{216} + \frac{1}{2}\frac{305}{1296}$ which amounts $\frac{671}{2592}$ or  3263 to 671.  This is about a 0.25 chance of death, compared to the soldier who's 12 to 2 is a one seventh, or 0.14 chance of death.  Twice as risky as the soldier suggested.  I'll leave it to you to watch the film to see how life turns out for the long nosed baboon.