Russian Roulette

I recently acquired the iRevolver app for my iPhone. It has done a wonderful job settling disputes and providing gambling action. Most recently, I used the app to settle a round of drinks with friends. Usually we play credit-card roulette, but this was a bit more novel. Here was the specifics of our game:

The revolver has six chambers. I loaded one bullet. Four of us randomly selected    the order of the game and each person has a choice when it is was his/her turn: 1) pull the trigger or 2) spin the cylinder and then pull the trigger.

I was the second player, "P2", and lost on turn number 6. Everybody spun the cylinder when it was his/her turn. Naturally, since our game allowed a strategic decision to be made (whether to spin or not spin the cylinder) I was curious whether one could use this decision to their advantage for future rounds.

A few things to note about this game:


1. Let the random variable X be the number of turns it takes until the single bullet is fired. Then, the Probability Density Function of X is

$PDF(x)=\frac{1}{6}*(\frac{5}{6})^{x-1}$ where $1 \leq x < \infty$ 

and 


$E[X]=\sum_{i=1}^{\infty}i*PDF(i)$


or more elegantly

$E[X] = \frac{1}{6}*1+\frac{5}{6}*(E[X]+1)$


so

$E[X] = 6$

2. $P(X>4) \cong \frac{1}{2}$. Since,

$P(X>4) = 1-P(X \leq 4) = 1-PDF(1)-PDF(2)-PDF(3)-PDF(4) = 0.48225$

These artifacts of the game made a one-bullet-per-cylinder game a fun choice for four people.


Back to answering the question about using the strategic decision to your advantage, I've created the below table which lists all possible bullet formations given a six cylinder chamber and the probabilities of losing when spinning vs not spinning the cylinder -- given you do not go first; therefore you are given new information about the bullet setup from the fact the player before you did not lose. [#_of_bullets, cylinder_setup, $P(lose\ |\ no\ information) = P(lose\ |\ spin)$, $P(lose\ |\ no\ spin\ \&\ P-1\ win)$):

0 (......) 0.00000 0.00000

1 (o.....) 0.16667 0.20000

  2 (oo....) 0.33333 0.25000 *

2 (o.o...) 0.33333 0.50000

2 (o..o..) 0.33333 0.50000

  3 (ooo...) 0.50000 0.33333 *

3 (o.oo..) 0.50000 0.66667

3 (o.o.o.) 0.50000 1.00000

  4 (oooo..) 0.66667 0.50000 *

4 (ooo.o.) 0.66667 1.00000

4 (oo.oo.) 0.66667 1.00000

5 (ooooo.) 0.83333 1.00000

6 (oooooo) 1.00000 1.00000

The asterix symbolize the instances where $P(lose\ |\ spin) > P(lose\ |\ no\ spin\ \&\ P-1\ win)$. So, there is an advantage to be had: when using two, three or four bullets set consecutively in the cylinder, don't spin when following a person who didn't lose.