**WARNING - Lots of maths follows, so scan down to the bottom if you don't care and just want the cliffnotes, also bolded**

I made a post a couple of months ago entitled Staying Alive, and I've now had the time to run some of the numbers about it. The gist of it was whether, on certain three dart outshots, it's better to go for the bull first to keep your chances alive into the second dart, or whether you have more chance of killing in the three dart combination as a whole.

Burton posted an analysis of Deller's 138 outshot against Bristow on his blog late last year, which gives us some of the figures we need. The bottom line is that van Gerwen, when asked, said he hits 25 or bull 80% of the time, and that in the European Tour events he was looking at, bull itself was hit between 25 and 30% of the time. I'm going to take the lower figure of those, simply through sanity checking - the bull is really quite small, so much so that I don't think that your average player is actually hitting it anywhere near the higher figure when compared to the amount of times they hit a much larger target, namely a double.

How often do players hit doubles and trebles? I've looked through the last World Championship final, and van Gerwen and Anderson, combined, were 44 out of 106 on doubles, or 41.5%. Looking through the dartsdata commentary, first dart in hand on scoring they were a combined 132 out of 293 on hitting a big treble, or 45%.

This is a strange situation - you would think that with doubles being larger than trebles, they would be hit more often, but there's a few counterarguments as to why the figure is lower. Firstly, it will incorporate darts at the bull, of which there were about one attempt per set before I got bored of counting. Secondly, with more than one dart in hand, players may be tempted to leave a marker rather than going straight at it, which will drag the percentage down. Thirdly, there's more pressure in going for a leg (in a world final as well) than there will be in your run of the mill hitting treble 20 to get there in the first place. But for now, we'll use those figures.

Three quick assumptions - we don't have any bounceouts, and we don't miss big numbers. The calculations have got to be somewhat quick to do. I'm also treating each dart as an independent variable - clearly they're not but it's something I need to do here. I'm also going to round off a bunch of figures where it's not going to make much difference for clarity purposes.

The staying alive way of hitting 135 is to go bull first. Let's start with worst case scenario - we miss entirely. 20% of the time this happens, and we're left with something in the 115-134 area to set up. If we assume that our misses are evenly distributed, then we have:

- Six cases (115-120) where we need just one treble to leave a shot at double. I know that on 119 if we hit the treble on the 19's we still need another one, but I'll count it like this as otherwise it's one sole case. 30% of the time we miss twice and are on a two darter in the 75-80 range, the rest of the time we are on a single darter.

- Fourteen cases where we need two trebles to leave a shot at double. 20% of the time we hit this. Same as before, 30% of the time we miss twice and are on something above 80 when we return, 50% of the time we miss once and hit once and have a single/double shot. We can improve a touch if we use the bull again, say when we are left with 121/125, but I'm grouping again as it's not going to add that much equity.

So overall we have 35% on a double, 35% on 41-60, 9% on 61-80 and 21% on 81-100.

Second scenario - we hit 25 first dart. Just the one permutation here which makes things easier. Now here we either hit treble 20 or we don't. We do this 45% of the time. If we do, 25% of the time we go out, 55% of the time we leave 25, and the rest of the time we leave something between 30 and 49 - six of these are shots at double, the rest aren't, so when we hit treble 20, it's 25% out, 6% on a double, the remaining 69% is fat number for double.

If we miss, then we're on 90. There's a few routes here but let's say we opt for treble 18. This is straight forward - 45% of the time we come back on 36, the rest on 72.

Overall, it's roughly 11% out, 27.5% on a double, 31% on big number for double and 30% needing 61-80.

Final scenario - we hit bull first to leave 85. We go for a big treble of choice and then either:

- Hit 45% of the time. Of that percentage, 41% of the time we go out, 59% we are still on a double when we return.

- Miss 55% of the time. This will leave us 70 or 66, depending on how much we like double 14 compared to tops. We then go for another treble, leaving a double if we hit or fat number for double if we don't.

Overall in this situation, it's about 18% out, 51% on a double, the rest needing a big number to be on a double.

Pull all that together, and the overall situation going bull first is:

Check out 10.8%

Leave double 34.92%

Leave fat number for double 31.64%

Leave 61-80 18.44%

Leave 81-100 4.2%

Got that? Good! Now let's go treble 19 first. 45% of the time we hit and leave 78. Much as I ignored some of the possibilities of maybe increasing equity by using bull, I'm going to ignore the possibility of going double-double to kill here, because I'm not Simon Whitlock, and the numbers say we hit trebles more often regardless, so it's a bad idea. Treble 18 it is. 45% of the time we hit again. This is about 20% of all cases. 24 left, and 41% there we go out. This is overall around an 8.3% chance of killing - which is less, but what happens when we don't go out is also key. Clearly the remaining 11+% is on a double, but what about the rest?

If we hit first dart and leave 78, but then miss treble 18, we will, using our assumption we don't miss big numbers, leave tops every time. This accounts for nearly one in four of all possible outcomes.

If we miss first dart and leave 116, we then hit a treble next next dart 45% of the time and leave a double every single time. This also accounts for nearly one in four of all possibilities.

If we miss first and second dart, then we're on 97 (or maybe 96, switching to 20's after missing first up is obviously fine) and need treble to leave a double. It's about one in three times where we miss twice, so hitting third dart adds another 13-14% of all cases where we're on a double if we return.

Finally, we can miss all three times, and leave something like 76, 78 - probability of this is a simple 55% cubed.

So our overall figures through NOT going bull first are:

Check out 8.3%

Leave double 75.06%

Leave 61-80 16.64%

That difference in leaving a double is huge. Enormous. I can't think that it's ever correct to try to check out an extra 1 in 40 legs in order to leave yourself more work to do 4 out of every ten legs should the opponent not kill. Not a chance. You'd have to be extremely sure your opponent will finish to give up the extra winning chances when you return.

I didn't pick 132 for a reason - firstly, going bull and hitting bull has more permutations, in that you can go bull again and it might make sense to do so, rather than bailing out to a treble as you have to do on 135. Secondly, if you opt for treble 20 first up, leaving 72 rather than 78 makes it a lot more enticing to try a double-double out - it's either two 18's or tops-double 16.

**Cliffs - yes, going bull first gets you out**

*slightly*more often in one visit. It also gets you out*significantly*less often in two visits.
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