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Russian Roulette (Posted on 2010-06-08) Difficulty: 4 of 5
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).

No Solution Yet Submitted by John zadeh    
Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution 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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