In the game of Russian roulette (not recommended by the author), one inserts a s

Question

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

Explanation / Answer

a) as probabilty of being shot in one shot =1/6

hence probabilty of being alive after one shot =1-1/6 =5/6

and all events of shooting are independent.

hence probabilty of being alive after N times =(5/6)N

b) probabilty of being alive till N-1 shoots and being shot at Nth time =(5/6)N-1 *(1/6)

c) let person is dead in Xth shot

hence expected number E(X) =1*probabilty of being shot in1 attempt +2*probabilty of being shot in2 attempts+ 3*probabilty of being shot in3 attempts+.................

=1*(1/6)+2*(1/6)(5/6)+3*(5/6)2(1/6)+.....

=1/(1/6) =6

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