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We were discussing a bet on Fred about two fat blokes throwing pointy things at a circle hung on a wall. They call it Prermier League Darts, & we discovered a method where we could not lose.
This was my summing-up.
Darts - probability of a draw in a 12 leg game.
I think that the probability of a draw in an even match (i.e. one where both players have the same probability of holding their own throw) is:
=(POWER(ProbWin;12)+36*POWER(ProbWin;10)*POWER((1-ProbWin);2)+225*POWER(ProbWin;8 )*POWER((1-ProbWin);4)+400*POWER(ProbWin;6)*POWER((1-ProbWin);6)+225*POWER(ProbWin;4)*POWER((1-ProbWin);8 )+36*POWER(ProbWin;2)*POWER((1-ProbWin);10)+POWER((1-ProbWin);12))
(in spreadsheet formula speak) where ProbWin is the probability of a player holding his own throw.
This gives the following figures:
Prob Holding Throw Prob Drawn Match
0.5 0.2256
0.55 0.2268
0.6 0.2307
0.65 0.2376
0.7 0.2484
0.75 0.2649
In any case, as discussed above, it assumes that the probability of holding your throw in a leg is independent of the current scoreline which seems unlikely.
This was my summing-up.
Darts - probability of a draw in a 12 leg game.
I think that the probability of a draw in an even match (i.e. one where both players have the same probability of holding their own throw) is:
=(POWER(ProbWin;12)+36*POWER(ProbWin;10)*POWER((1-ProbWin);2)+225*POWER(ProbWin;8 )*POWER((1-ProbWin);4)+400*POWER(ProbWin;6)*POWER((1-ProbWin);6)+225*POWER(ProbWin;4)*POWER((1-ProbWin);8 )+36*POWER(ProbWin;2)*POWER((1-ProbWin);10)+POWER((1-ProbWin);12))
(in spreadsheet formula speak) where ProbWin is the probability of a player holding his own throw.
This gives the following figures:
Prob Holding Throw Prob Drawn Match
0.5 0.2256
0.55 0.2268
0.6 0.2307
0.65 0.2376
0.7 0.2484
0.75 0.2649
In any case, as discussed above, it assumes that the probability of holding your throw in a leg is independent of the current scoreline which seems unlikely.
I've just compared against our model and you(or whoever you stole the quote from isn't far off). You'd rarely get exactly the same hold %'s which is a factor, along with momentum. The ad break in Premier League darts used to be a big momentum killer, though it doesn't exist anymore so the players struggle to buck the trend now. We actually feel like the draw price is far too big in the market, just doesn't seem to be mathematically correct. However in a world where majority just copy the prices it doesn't look to change, so the draw is often a smidge of value on a Thursday.
Well worked out Tikay!
