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 Russian Roulette (Posted on 2010-06-08)
Eight men, including Colonel Mustard, sit at a round table, for a modified game of Russian roulette. They are using a six chamber revolver which has been loaded with 5 bullets.

The game begins by one of the men reaching into a hat, and randomly drawing the name of the first player.

If the first player survives his turn, the gun is handed to his adjacent clockwise neighbor, and his name is immediately returned to the hat.

If the first player loses, his name is thrown away, and the men pull from the hat, and choose the name of the next player.

The game is continued in such a way until either all five bullets have fired, OR a player survives his turn, but no longer has an adjacent clockwise neighbor to pass the gun to.

What is the probability that the Colonel will survive the game?

(Note that the chamber is spun every time a player takes his turn).

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 Exact solution | Comment 9 of 12 |
(In reply to "Exact" solution by Charlie)

Translating the program into UBASIC allows the exact solution:

`  300    dim Man(7),DeadProb(5)  400    500    for I=1 to 7:Man(I)=1:next  600    Dead=1:Bullets=4:MenRemain=7  700    CurrProb=1  800    900    gosub *ChooseNext 1000   1100   1200    for I=1 to 5:print DeadProb(I);:next:print 1300    T=0 1400    for I=1 to 5 1500      T=T+I*DeadProb(I) 1600    next 1605   print:print:print 1610    for I=1 to 5:print DeadProb(I)/1;:next:print 1700    print T,T//8,1-T//8:print T/1,T/8,1-T/8 1701   print CtA,CtB 1710   end 1800   1900    *ChooseNext 1910      local MenBefore,M,I,TotSubProb,SubProb,PDie 2000      MenBefore=0 2100      for M=1 to 7 2200       if Man(M)=1 then 2300   :TotSubProb=0 2400   :SubProb=1//MenRemain 2500   :for I=1 to MenBefore 2600   :TotSubProb=TotSubProb+SubProb*((6-Bullets)//6)^(MenBefore+1-I) 2700   :next 2800   :PDie=(TotSubProb+SubProb)*Bullets//6 2900   :' dies: 3000   :Dead=Dead+1 3001   : 3100   :CurrProb=CurrProb*PDie 3200   :Man(M)=0 3300   :Bullets=Bullets-1 3400   :MenRemain=MenRemain-1 3500   : 3600   :if Bullets>0 then 3700   :Lvl=Lvl+1:gosub *ChooseNext:Lvl=Lvl-1 3800   :else 3900   :DeadProb(Dead)=DeadProb(Dead)+CurrProb 3901   :CtA=CtA+1 4000   :endif 4100   : 4200   :MenRemain=MenRemain+1 4300   :Bullets=Bullets+1 4400   :Man(M)=1 4500   :CurrProb=CurrProb//PDie 4600   :Dead=Dead-1 4700   : 4800   :' doesn't die: 4900   :if M=7 then Cntnu=0:else if Man(M+1)=0 then Cntnu=0:else Cntnu=1:endif:endif 5000   :if Cntnu=0 then 5100   :DeadProb(Dead)=DeadProb(Dead)+CurrProb*(TotSubProb+SubProb)*(6-Bullets)//6 5101   :CtB=CtB+1 5200   :endif 5300   : 5400   : 5500   :MenBefore=MenBefore+1 5600   :else 5800   :MenBefore=0 5900   :endif 6000   6100      next 6200    return 6300  `

finds that the respective probabilities of 1 through 5 deaths are:

1093/15309, 110183/489888, 308637139/892820880, 251655251035/925676688384, 398693251609/4628383441920

or approximately

0.0713959109020837415, 0.2249146743745509177, 0.345687635575906334, 0.2718608496821143168, 0.0861409294653446896

The exptected number of deaths is 14238926425753/4628383441920, making the probability of a given man dying 14238926425753/37027067535360, or of surviving, 22788141109607/37027067535360, including the colonel.

The respective decimal approximations of these three numbers are given as:

3.0764362124340852952   0.3845545265542606618   0.6154454734457393381.

 Posted by Charlie on 2010-06-09 14:22:27

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