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).
(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.
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Posted by Charlie
on 2010-06-09 14:22:27 |
Exact solution
