Russian roulette with a gun that spins randomly. What are the chances of never being shot and how to express the diminishing chances of that outcome

probabilityrandom

My math is bad but I'm intrigued by a problem.

I was part of this discussion earlier today. In an imaginary Russian Roulette game, there's the standard 6 holes and single bullet in the gun, but once the trigger is pressed, the cylinder's spin is random. Both the direction and amount of holes it will move are random.

Some people where adamant in claiming you can press that trigger a billion times and not get shot. That seems to me, impossible, either in principle or in practice. Now see, the points I stood for were:

  1. It seems impossible, in theory, for that bullet not to
    ever be fired, because if the cylinder's spin is truly random, it
    will eventually land on every single hole, on average, 1 out of
    every 6 times. I'm saying this based purely on the intuition that
    given such a small set, an algorithm avoiding one or more specific
    elements expresses a predictable pattern and can't be truly random.
  2. If I'm wrong and it's theoretically possible for the bullet to
    never be fired, it feels that in practice it's not. It seems to me that the chances the bullet will not be fired
    are inversely proportional to the number of trigger pulls. That is,
    if the trigger is never pulled, there's a 100% chance the bullet
    won't be fired. If if it's pulled once, that certainty has to
    go down. Given enough trigger pulls then it goes down enough to
    reach and then cross the threshold under which it's a
    positive integer. Then either it becomes a decimal smaller than 1 or a
    negative integer, at some point that value will no longer be relevant and one
    can assert the bullet has been fired.

So I have 2 problems here that you can help me with

  1. I'm bad at math and can't prove either point
  2. If either one is provable, I don't know what's the way to express it concisely. The simplest explanation I came up with is the trigger pull x certainty relation I came up with.

EDIT:
I''d like to thank you all who answered and commented. This had left me super intrigued and excited, and was oersonally very important to me, because even if I used to be one of the worst in class for math, it showed me that given an interesting wrapping and enough logical thinking, one can get far in solving such a problem.

I wish one day I'll adult.life will guve me enough of a break so that I have time to priperly learn math, the thing is amazing.

Best Answer

Pro(nevrr)=$(\frac{5}{6})^n$ for n tries. As you see it gets smaller as n increases.

Powers that be will kick you off.

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