Dueling on the Cheap

You have one six-shooter, one bullet, and two offended fools, determined to duel out their differences. First one will spin the cylinder (placing the bullet in a random location of six in the cylinder) and then shoot at the other, at point-blank range. If the gun does not fire, the second duelist will spin the cylinder, and then fire back at the first person. If the gun still does not fire, the first person will again spin and fire, and so on.

How important is it to go first? How long will the duel last?

The sample space contains all possible outcomes.

a = gun fires for person A
b = gun fires for person B
x = gun does not fire

The duel lasts for k pulls of the trigger, and the gun does not fire for any of the first k - 1 pulls but does fire on the kth pull. The sample points for which A wins are highlighted in red (all of the sample points of odd length).

a
x b
x x a
x x x b
x x x x a
x x x x x b

The probability of A winning the duel can be expressed as follows.

Person A thus has slightly better odds of winning than does person B.

We now define D as the duration of the duel (in units of trigger pulls). The average, or expected, value of D is then

Consider the following revision of the rules, if person B complains that he or she has an unfair disadvantage. Suppose that A pulls the trigger once, then B pulls the trigger twice (always spinning the cylinder to a random position between firings), then A pulls the trigger three times, then B four times, and so forth.