Thank you for correcting my error. My error was that I just considered
the possibility of a win or a loss in the first game. I forgot to
calculate what happens if the first game is a draw.
I took a nap and in my sleep realized that I had made an error.
I also calculatre that if we split the first two games 1-1, then my
odds of winning the match are exactly 50%. This is because I need only
one win or two draws in the last two games.
Now, what I left out was:
DWW
DWD
DWLW
DWLD
DWLL
DDW
DDDW
DDDD
DDDL
DDLW
DDLD
DDLL
DLWW
DLWD
DLWL
DLDW
DLDD
DLDL
DLL
There are 19 possibilities above. I win 11 and lose 8, but since L
occures twice as often as D or W the odds are slightly against me.
Sam Sloan
On 24 Jun 2005 19:07:32 -0700, Paul Rubin
wrote:
(Sam Sloan) writes:
Now, add them all up and you get Brock's chamces of winning the four
game match.
That is actually a reasonable approach, but there's more cases to
consider. I think the following is a complete list and the final
result agrees with Kane. This is based on 25% draw probability
(constant for each game) and having the white pieces is worth 50
points. Note that if Sloan gets white in the first round, his chances
of winning the match are about 45%, while if he gets black, his
chances are only about 20.5%. This match will possibly be decided by
the color draw at the beginning, assuming colors are chosen that way.
================================================= ===============
Win / Lose / Draw probability:
Sloan white: W=0.288105 L=0.461895 D=0.250000
Sloan black: W=0.158576 L=0.591424 D=0.250000
*** Sloan gets white in first round
WW--: Sloan wins, prob=0.083004
WLW-: Sloan wins, prob=0.038339
WLLW: Sloan wins, prob=0.017709
WLLL: Brock wins, prob=0.028391
WLLD: Brock wins, prob=0.015367
WLDW: Sloan wins, prob=0.009585
WLDL: Brock wins, prob=0.015367
WLDD: Sloan wins, prob=0.008317
WDW-: Sloan wins, prob=0.020751
WDLW: Sloan wins, prob=0.009585
WDLL: Brock wins, prob=0.015367
WDLD: Sloan wins, prob=0.008317
WDD-: Sloan wins, prob=0.018007
LWW-: Sloan wins, prob=0.038339
LWLW: Sloan wins, prob=0.017709
LWLL: Brock wins, prob=0.028391
LWLD: Brock wins, prob=0.015367
LWDW: Sloan wins, prob=0.009585
LWDL: Brock wins, prob=0.015367
LWDD: Sloan wins, prob=0.008317
LLWW: Sloan wins, prob=0.017709
LLWL: Brock wins, prob=0.028391
LLWD: Brock wins, prob=0.015367
LLL-: Brock wins, prob=0.098544
LLD-: Brock wins, prob=0.053337
LDWW: Sloan wins, prob=0.009585
LDWL: Brock wins, prob=0.015367
LDWD: Sloan wins, prob=0.008317
LDL-: Brock wins, prob=0.053337
LDDW: Sloan wins, prob=0.008317
LDDL: Brock wins, prob=0.013334
LDDD: Brock wins, prob=0.007217
DWW-: Sloan wins, prob=0.020751
DWLW: Sloan wins, prob=0.009585
DWLL: Brock wins, prob=0.015367
DWLD: Sloan wins, prob=0.008317
DWD-: Sloan wins, prob=0.018007
DLWW: Sloan wins, prob=0.009585
DLWL: Brock wins, prob=0.015367
DLWD: Sloan wins, prob=0.008317
DLL-: Brock wins, prob=0.053337
DLDW: Sloan wins, prob=0.008317
DLDL: Brock wins, prob=0.013334
DLDD: Brock wins, prob=0.007217
DDW-: Sloan wins, prob=0.018007
DDLW: Sloan wins, prob=0.008317
DDLL: Brock wins, prob=0.013334
DDLD: Brock wins, prob=0.007217
DDDW: Sloan wins, prob=0.004502
DDDL: Brock wins, prob=0.007217
DDDD: Sloan wins, prob=0.003906
Match probabilities: {'Sloan': 0.44910257290666628, 'Brock': 0.55089742709333378}
*** Sloan gets black in first round
WW--: Sloan wins, prob=0.025146
WLW-: Sloan wins, prob=0.014872
WLLW: Sloan wins, prob=0.008796
WLLL: Brock wins, prob=0.032805
WLLD: Brock wins, prob=0.013867
WLDW: Sloan wins, prob=0.003718
WLDL: Brock wins, prob=0.013867
WLDD: Sloan wins, prob=0.005862
WDW-: Sloan wins, prob=0.006287
WDLW: Sloan wins, prob=0.003718
WDLL: Brock wins, prob=0.013867
WDLD: Sloan wins, prob=0.005862
WDD-: Sloan wins, prob=0.009911
LWW-: Sloan wins, prob=0.014872
LWLW: Sloan wins, prob=0.008796
LWLL: Brock wins, prob=0.032805
LWLD: Brock wins, prob=0.013867
LWDW: Sloan wins, prob=0.003718
LWDL: Brock wins, prob=0.013867
LWDD: Sloan wins, prob=0.005862
LLWW: Sloan wins, prob=0.008796
LLWL: Brock wins, prob=0.032805
LLWD: Brock wins, prob=0.013867
LLL-: Brock wins, prob=0.206869
LLD-: Brock wins, prob=0.087445
LDWW: Sloan wins, prob=0.003718
LDWL: Brock wins, prob=0.013867
LDWD: Sloan wins, prob=0.005862
LDL-: Brock wins, prob=0.087445
LDDW: Sloan wins, prob=0.005862
LDDL: Brock wins, prob=0.021861
LDDD: Brock wins, prob=0.009241
DWW-: Sloan wins, prob=0.006287
DWLW: Sloan wins, prob=0.003718
DWLL: Brock wins, prob=0.013867
DWLD: Sloan wins, prob=0.005862
DWD-: Sloan wins, prob=0.009911
DLWW: Sloan wins, prob=0.003718
DLWL: Brock wins, prob=0.013867
DLWD: Sloan wins, prob=0.005862
DLL-: Brock wins, prob=0.087445
DLDW: Sloan wins, prob=0.005862
DLDL: Brock wins, prob=0.021861
DLDD: Brock wins, prob=0.009241
DDW-: Sloan wins, prob=0.009911
DDLW: Sloan wins, prob=0.005862
DDLL: Brock wins, prob=0.021861
DDLD: Brock wins, prob=0.009241
DDDW: Sloan wins, prob=0.002478
DDDL: Brock wins, prob=0.009241
DDDD: Sloan wins, prob=0.003906
Match probabilities: {'Sloan': 0.20503126482804437, 'Brock': 0.79496873517195532}
Total probabilities: {'Sloan': 0.32706691886735534, 'Brock': 0.67293308113264461}
|