In the game of Russian roulette (not recommended by the...

In the game of Russian roulette (not recommended by the author), one inserts a single cartridge into the drum of s revolver, leaving the other five chambers of the drum empty. One then spins the drum, aims st one's head, and pulls the trigger. (a) What is the probability of being still alive after playing the game $N$ times? (b) What is the probability of surviving $(N-1)$ turns in this game and then being shot the $N$ th time one pulls the trigger? (c) What is the mean number of times a player gets the opportunity of pulling the trigger in this macabre game?

In the game of Russian roulette (not recommended by the author), one inserts a single cartridge into the drum of s revolver, leaving the other five chambers of the drum empty. One then spins the drum, aims st one's head, and pulls the trigger.
(a) What is the probability of being still alive after playing the game $N$ times?
(b) What is the probability of surviving $(N-1)$ turns in this game and then being shot the $N$ th time one pulls the trigger?
(c) What is the mean number of times a player gets the opportunity of pulling the trigger in this macabre game?

Key Concepts

Independent Trials

Each round of the game can be viewed as an independent trial because the outcome of each pull of the trigger does not affect the outcomes of future pulls. In probability theory, independent events allow the computation of the joint probability of a sequence of events by multiplying the individual probabilities.

Multiplicative Probability

This concept refers to the method of calculating the probability of a sequence of independent events by taking the product of the individual event probabilities. It is used to determine the probability of surviving multiple rounds (by multiplying the survival probability for each round) or a specific sequence where survival is followed by a fatal outcome.

Geometric Distribution

The situation described in the game fits the framework of a geometric distribution, which models the number of trials until the first occurrence of a success (in this case, being shot). This distribution is characterized by a constant probability of success in each trial and is key for understanding the pattern of first occurrence events.

Expected Value in Geometric Distribution

One important property of the geometric distribution is that its mean, or expected number of trials until the first success, is the reciprocal of the success probability. This provides a concise way to compute the average number of trigger pulls a player might experience before being shot, encapsulating the idea of waiting time in random processes.

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