The Deller: A Statistical look at whether Bristow should have gone for the bull

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December 2, 2016 by bsd987

[3 Dec 2016 UPDATE: Wayne Mardle has asked Michael van Gerwen how often he thinks he misses the bullseye or 25 entirely when aiming at bull. He estimated that he would miss the bullseye or 25 about 20 times out of 100. By comparison, Vincent van der Voort estimated 25 times out of 100 for himself. I have updated the last section of this article with van Gerwen’s estimate of 20 times out of 100, as I do not have the exact numbers to use. The original version of this article used the number as 12 out of 100.]

A year ago, when I started collecting stats, I asked the question that has rattled the darts world more than three decades: Did Eric Bristow mess up by not throwing at bull and thus giving Keith Deller a shot to check out 138 for the 1983 Embassy World Championship title?

Now, my statistics are only based on 2016: In 1983, the equipment was inferior to what the players have today, including both the darts and boards. Bounce-outs were more frequent, regardless whether we are talking on the trebles, doubles, or bulls. It’s impossible to know what the exact percentage shot a 138 would have been in 1983. But let’s pretend that we’re not in 1983 but rather we’re in 2016. And let’s replace both Eric Bristow and Keith Deller with the ‘average’ PDC stage player (which, of course, is actually an above-average PDC player, since those that qualify for the stage events tend to be the above-average players).[1] The question is simple: How often would the average PDC player win the leg going for bull versus going for S18?

First, we need to know how often the bull is actually hit. There’s a bit of conflicting data for this. I’ve taken a look at the bullseye hit rate both before and after the switch to the Unicorn HD2 boards, and although it’s still a small sample, the bullseye rate has gone down, at least on the European Tour. In the last 4 Euro Tour events with the HD1 board, the bull was hit 30.52% of the time. In the 4 Euro Tour events since the switch, it’s only been hit 26.81% of the time.[2] I’ve ran the numbers both ways, giving us two base probabilities: A minimum using the 26.81% number, and a maximum using the 30.52% number.

Second, we need to know the likelihood that Deller would hit the 138. If he’s an average player in 2016, he would hit the 138 approximately 5.71% of the time, or slightly more often than once in 20 attempts.

Third, we need to know the likelihood of Bristow checking out 32 next visit. Assuming he hits the big number 100 percent of the time—which is not a certainty, but is close enough that I’m going to pretend it is since I don’t have the data as to how often he would miss the big number—Bristow would come back for a 72.31% outshot: The average 2016 player would take out 32 in no more than 3 darts a little less than three-quarters of the time.

Fourth, we would need to know what Deller has left, assuming he misses the 138. I’ve run the numbers on all realistic possibilities: That he’s left a double, that he’s left a single-double outshot, and that he’s left a finish of around 80. While there’s also the possibility that he’s left something above 80, at that point we’re getting into situations where either way Bristow would win the leg so often as to not be interesting to discuss. For reference, Deller would hit with 3-in-hand 72.31% of the time, on a single-double out 61.19% of the time, and on a more complex out under 80 44.99% of the time.

Now, onto what would happen in each scenario.

If we use the 26.81% bull number, this would mean that 26.81% of legs should end before Deller even got a dart at the board. We’ll put this in the Bristow column. Bristow wins 26.81% of the time; Deller gets a shot (but does not necessarily win) 73.19% of the time.

When Deller comes to the board, he should be expected to check out 5.72% of the time. But when multiplied by the 73.19% of the time he gets a shot, he wins 4.18% of the time. Bristow would come back the other 69.01% of the time for what would likely be a two-dart out.[3] A single-double out is hit 61.19% of the time, meaning Bristow wins this visit 42.22% of the time.

Now we reach the difficulty: If Deller did not hit the 138, what did he leave? I’ve split the calculation out into 3 different scenarios: Deller left 40, Deller left 50, and Deller left 78. These take us into the 3 most likely types of outshot he has left. I’ve run the calculus for all three, both assuming Bristow throws at bull and assuming he does not. Here are the chances of Bristow winning the leg in each scenario.

Bull S18 Difference
Deller Leaves 40 74.84% 73.83% 1.01%
Deller Leaves 50 77.17% 76.11% 1.06%
Deller Leaves 78 80.57% 79.42% 1.15%

As you can see, in each situation, going for bull increases Bristow’s chance of winning the leg by about 1 percent, a fairly insignificant chance. Yet, remember, we used some favourable numbers and some unfavourable numbers: We used a bull hit rate that excluded all darts that did not hit either the 25 or the bull; but we also used the lower hit rate of 26.81%, not the older rate of 30.52%. Let’s compare when we change the bull hit rate to the even more favourable 30.52%.

Bull S18 Difference
Deller Leaves 40 76.12% 73.83% 2.28%
Deller Leaves 50 78.33% 76.11% 2.22%
Deller Leaves 78 81.55% 79.42% 2.13%

Using ALL favourable assumptions towards going for bull, the bull becomes the percentage play, but only slightly. It increases the chances of winning the leg by about 2%. But we know players miss the 25 entirely sometimes, and the real bull percentage hit rate is probably somewhere in between the two numbers. At worst, using 2016 data, Bristow made a very small statistical error by going for 18. At best, he made a choice between two relatively even plays.

But we’ve made another assumption that isn’t quite accurate: Eric Bristow is not the average player. While Deller in 1983 could have passed for average: He was unheralded and while he had a good run at Lakeside, his performances didn’t quite stick out from the rest of the field. And the rest of his BDO career was far from spectacular: He was even knocked out at Lakeside first round the following year by Nicky Virachkul, the man he beat in the first round in 1983. Bristow, of course, won the next three world titles, and would play in seven of the next eight Embassy finals.

So let’s change the numbers: Let’s make Eric Bristow not the average professional but rather Michael van Gerwen. I’ve imported MvG’s 2016 statistics, on both finishes and on hitting the bullseye. I’ve kept Deller the average professional.

Bull S18 Difference
Deller Leaves 40 78.17% 77.89% 0.28%
Deller Leaves 50 80.29% 79.84% 0.45%
Deller Leaves 78 83.39% 82.68% 0.71%

Using MvG’s numbers as Bristow’s numbers, the odds of Bristow winning the leg are almost the same either way. There is still a slight, slight probabilistic favourability to going for bull. But remember, I’ve used numbers that assume that when MvG throws at the bullseye, he either hits the 25 or the bullseye. I do not have numbers for his exact hit rate, including darts that miss the 25 entirely or bounce out.

So let’s make two more calculation, this time assuming MvG would miss the bullseye or 25 entirely 20% of the time, which is van Gerwen’s personal estimate of how often he would miss the bull.[4]

Bull S18 Difference
Deller Leaves 40 76.25% 77.89% -1.64%
Deller Leaves 50 78.56% 79.84% -1.28%
Deller Leaves 78 81.93% 82.68% -0.75%

Using these numbers, the numbers tilt slightly towards going S18, making it Bristow more likely to win the leg about 1 time out of 100. But yet again, it’s extremely slight.

What can we conclude? In 2016, the two choices, statistically speaking, are roughly equal. Hindsight may say Bristow messed up. But in reality, he chose between two roughly equal decisions and lost out.

[1] Later, I’ll replace the average player with Michael van Gerwen.

[2] This actually excludes attempts that miss both the bull and the 25 altogether. However, I’ll (incorrectly) assume that that never happens, since it would only decrease the likelihood of hitting the bull and thus depress the numbers for the counterfactual scenario of Bristow going for the bull.

[3] I recognize that there would be the slim chance that Bristow missed the 25 entirely and instead hit 10, 12, 14, 16, 18, or 20, thus leaving a simpler 3-at-a-double finish. Because this happens so infrequently, I’m assuming it never happens. Its effect on the actual calculation, while real, would be minimal.

[4] The original version of this article estimated at 12%, which was a compromise number after I, Rod Studd, and Wayne Mardle all suggested different numbers. Special thanks to both for consultation on this article, as well as to Mardle for asking Michael van Gerwen for his estimate of how often he would miss the bull or 25 entirely.

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