Darts mathematics, probabilities, and out shots
What is independence?

Two events are independent, if one doesn't effect the other. There are many popular examples, I'll stick with dice rolls here.

Dice rolls are independent. That is, if you roll a die, the probability for a given number is 1/6. Always. Roll it a thousand times, it's still 1/6th. And it doesn't matter what the last roll was. That is "independence". Even if the last three rolls were "6-6-6", the probability for another "6" is just again 1/6.

That doesn't mean that rolling "6-6-6" is easy. In fact, the probability to do so is 1/6 * 1/6 * 1/6 = 1/216. That's about 0.46 per cent.

But: After rolling "6-6" with the first two, the probability for totaling a "6-6-6" is 1/6. Which should be obvious, as the die has no brain, and just shows a random number again with probability 1/6.

Are dart throws independent?

Now let's get on to the intersting things. Darts. Much more fun.

Are the three darts independent of each other? Well, yes with limitations. That's get the limitations out first. There are things like "muscle memory", "follow up", "blocking dart" etc, but this is not what we want to look at here.

Consider throwing three darts at three different targets. Say S16, S20, S10. Are they dependent on each other? "No, of course not" you say, and you're right. You do this a hundred times, and you get all variations of hitting or missing the targets. Hitting S16 has no influence on hitting S20 or S10. Your darts have no connection to each other. Having hit one target - the next dart doesn't care. Think throwing with light off, where you cannot even see if you hit. It should be very obvious now that one throw is as good as the next one.

Let p(S_hit/S_aim) or short p(Sh/Sa) the probability for hitting the single when aiming at the single.

If you did that experiment for quite a number of rounds, you get a good feel for your hitting probability, like 4/5.

Now we can take this further to other targets, i.e. trebles and doubles. There are probabilities for all combinations of targets and hits, but we will stick to the simpler ones.

Does it matter if you swap an easy target (i.e. S16) for a harder one, i.e. (T16)? Your hit probability for the T16 will be lower than for the S16, but throwing your other darts on S20 and S10 won't change. Remember: You could always throw in the dark. There is no way the darts know if the previous darts have hit their target.

Ok, so all three darts are independent.

All cool - but what do the probabilities mean?

Just for the sake of it, let's try some values here...

First example: Aiming for treble, hitting it one in three, and for simplicity two of three go to the corresponding single.

p(T/T) = 1/3
p(S/T) = 2/3

E(T) = 1/3 * 3 + 2/3 * 1 = 5/3
3 * E(T) = 3 * 5 /3 = 5

E is what to expect, i.e. your three darts will (no surprise) score 5 marks, i.e. 100 points on average on T20. That's 166 marks for the 100 darts @ 20 game. Pretty impressive.

Let's go with 1/4 and 3/4 instead.

p(T/T) = 1/4
p(S/T) = 3/4

E(T) = 1/4 * 3 + 3/4 * 1 = 6/4

That's an average of 90, or 150 marks for the 100 darts @ 20 game. Still excellent.

So reality for most amateurs is well below that, but then there are those neighbor fields as well.

So what's the probability for a 180 then?

Well, in theory you just take p(T/T) * p(T/T) * p(T/T), giving 1 in 27 for 1/3, or 1 in 64 for 1/64. Better players will take advantage of follow up, stacking etc., but deflection will make some of this void.

But it's always the third dart that fails!

Well, no. This is a typical mind game. Yes, deflection is effecting your third dart the most, but often the third dart doesn't touch the others at all. The thing here is that our brain will see two trebles and think "whoa, almost a one eighty", and then the last dart follows. You just don't think that after hitting S20 first, even if you follow up two T20s. In practice, one is as good as the other, yet one seems to be so much closer than the other.

Don't let the brain fool you. Don't throw away the third dart after being disgusted by the first one. Each one counts on its own!

But that indepence thing - what's the point?

Yeah, right. Get that into your brain as quick as possible. There is no dependency of your darts. The next dart you throw is just another dart and will hit or not hit, no matter of your previous one.

Why is that important? Because it makes your mind free. Think 170 is hard? It sure is! But after having hit T20 T20, it's just you and the bull. It's not any more difficult than going bull after messing up 70 with S1 S19. Or your last dart for 89 after S19 S20.

It's a dart at bull. You will it with some probability. Your previous darts? Don't matter at all.

But wait - 170 is so much harder to take out than 89. Yes, it is, but not because the bull is any harder, but you need those two trebles up front. As soon as you are down to the bull, those are just reality.

It's as easy as that - just like in the dice example above. After "6-6" have been rolled, the great "6-6-6" combination is just as likely as any 6 with any random throw.


90 - 130 - 170

Those are great finishes to compare the probabilities.
90 doesn't need a treble, but benefits from one (giving one or two shots at a big double).
130 needs as least one treble, but benefits from two trebles (giving one dart at a big double).
170 needs two trebles and the bull.
Just remember: If left on the bull for the third dart - a 90 is just as hard as a 170. One dart, one try. Period.


Why do you tell us all this?

This post was inspired by Cyanides pretty good outshout thread - in fact one of the best that's currently out there.

But at one point, he mentioned something like "When you hit T19 on a 129, you are less likely to finish that if you had hit S19". His argumentation is that hitting S19 T20 Bull would be easier than hitting T19 T20 D6. While the latter may or may not be right depending on all your probabilities on bull, trebles and doubles, it totally ignores the fact of indepence.

That is, as soon as the first dart has left your hand, it's gone. And as the darts are independent of each other, the next two darts are not impressed by the great hit of the first one.

The second dart can be thrown at T20 no matter what. And it will hit or not hit, totally independent of your first dart. It will not say "ah, you just got a treble, than I better do not, because you're just an average player". It just goes in the treble with the same probability as always. It's more likely to fail than to succeed. Because that's what it is. You hit maybe one in four (remember that light-off thing).

So let's assume you got lucky and it hit. What's left was only determined by your first dart. It can be 12 or 50.

One dart in hand. One target to go. If you're a typical darts player, your probability hitting the bull will be much lower than hitting a full sized double.

Remember - that one dart is totally independent from your other two darts. That dart just doesn't care about them.

I hope it's clear now why hitting T19 is much better than hitting S19. Your second dart is totally irrelevant to that finish. Hit T20 or you're lost. But getting lucky on the first one and hitting treble will leave a big double, while a single will just leave that little red center.

Bottom line

It's about the score left, and the number of darts in your hand.

Totally irrelevant is how you get there. Dart on the floor? Miscalculation? Great setup dart? Your remaning darts couldn't care less!


Other implications

This independence has some implications on the order of throw - or the lack thereof. As the probabilities are just multiplied, order doesn't matter. That's why you can throw a dart at S/T18 any time when going from 305. The two T20s (or T20 and T19) are "must hits". Probability for finish is zero if you miss them. Then there's one dart to avoid the boogie. p("have finish") = p(T20) * p(T20) * p("any 18"). Targets are mutually exclusive, you won't hit one when aiming for the other.

Generally, order does matter though.

I think there has been some change recently among the professional players - they seem to be stuck on the 20s now and prefer to use the last dart to get things right.

Provocative thesis

A funny thing with that 129 finish is: Everyone tells you to start on 19. I do it, too. But in this particular case it's irrelevant, because that T20 is a must hit. It's more or less a mental thing here - you keep the hope for the finish alive as long as possible. Mathematically it's the exactly same thing - it boils down to hitting that T20 when you throw at it. But you are just as likely to hit it with your first dart as you are with your 2nd - there is just no room for error. I wouldn't be surprised to see more players going T20 first in the next years.

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And remember class, there will be a test on this tomorrow morning so be prepared.
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(11-19-2014, 11:42 PM)ico Wrote: I wouldn't be surprised to see more players going T20 first in the next years.

That pretty much sums up everything you typed right there.
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i know all about probability, unfortunately ill probably miss.
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Here is the problem...

Darts isn't a pair of dice.

these guys practice 170 all the time. They hit them all the time. The transition from one target to another SHOULD be exactly the same. The targets SHOULD be irrelevan. Here's the thing.

what target do they aim at more than 90% of the time? T20.

I'm not saying anyone is right or wrong on this whole 129 thing... I'm saying your analysis isn't taking the human element into account.

give me 2 darts. Tell me to hit T20 with one, and T12 with the other. I can promise you I have a much better chance of hitting that t20.
-Milky

Keeping dart retailers in business since 2012.
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(11-20-2014, 01:56 AM)Milkysunshine Wrote: Here is the problem...

Darts isn't a pair of dice.

these guys practice 170 all the time. They hit them all the time. The transition from one target to another SHOULD be exactly the same. The targets SHOULD be irrelevan. Here's the thing.

what target do they aim at more than 90% of the time? T20.

I'm not saying anyone is right or wrong on this whole 129 thing... I'm saying your analysis isn't taking the human element into account.

give me 2 darts. Tell me to hit T20 with one, and T12 with the other. I can promise you I have a much better chance of hitting that t20.

Nailed it!
Play like it means nothing when it means everything


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Your main point seems to be that you are much more likely to finish 129 on bull than on D6. By all means, this is right.

Say you're hitting trebles one in four (being a 90 average), big doubles one in four, and bull one in six.

You throw your first dart at T19, your second at T20.

Now you are taking 1000 tries at 129, of which you will finish it on the bull 31 times, but only 16 times on D6.

Here is the "but". After hitting the T19, you have a 6 % chance of winning, after hitting S19 it's only 4%.

The reason for the huge lead of the bull is that you will throw at it three times at it for every dart you get on D6.



Did you understand the dice, or do you object to that as well? Is rolling a "6" any harder after rolling "6-6" than after rolling "3-2"? Serious question for you - it is the same thing. The previous throws/rolls have already happened. They will not influence the future.

Can you explain how the outcome of one dart can effect another, even if you wouldn't know it? What if the first dart falls out while the third dart is already thrown - will it effect its probability of hitting?

Not sure how should explain this even more, if you don't care to answer those questions. Your wrong assumptions seem to be put in stone, and you don't even allow for any doubt. Take it to any math teacher or professor or whoever you trust.

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You have explained your point very well ico. It's a good post about the maths side of darts. But each dart isn't really independent as we are human. If robots were throwing darts with high accuracy then possibly but darts is a highly mental, personal game.
Also I think the probability of getting the bull after the two triple 20s in your 170 example (or the third 6 in the 6*6*6 dice throw) isn't the same as you can't rule out the previous two throws. Even if each throw of the dice is always 1/6 a darter does have the own mind. Confidence and personal experience mixed with nerves then you can't deny the previous two throws. Independence and probability in mathematics or dice throw is one thing but the mind of a darter is another.
Darts and they're relationship with the thrower can't be denied and even the best players can't completely empty they're mind to put the each dart is independent theory into full practice.
But this is only my opinion
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Ok, here's what you're saying (or what that implies)

1) On 126, if you are hitting two T19s instead of just one. Now you're on D6 instead of bull. You are saying that you are less likely to take out D6 than bull with your last dart.

2) No reason to make the darts memorize just the previous one or two darts. If you're left on D20 with three darts in hand, you are less likely to take them out if your last throw was 180 instead of 26.

3) You are left with 50. Throw your first two darts at the floor, then go for the bull. It's easier than going S10 D20. This would need three marks instead of two, and hitting single is clearly more challenging than hitting the floor, even my dead granny would agree.

4) Throw 1000 rounds: First dart T9, second T7, third T3. As you will hit some trebles with your first darts, you will have much less hits on T3 than on the others. If you reverse the order of throw, you will have much less hits on T9 now.



Serious - if you think that one of this is totally wrong, then your other assumptions are as well. Because this is just following the very same "logic".

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(11-20-2014, 07:46 AM)JayKay Wrote: You have explained your point very well ico. It's a good post about the maths side of darts. But each dart isn't really independent as we are human. If robots were throwing darts with high accuracy then possibly but darts is a highly mental, personal game.
Also I think the probability of getting the bull after the two triple 20s in your 170 example (or the third 6 in the 6*6*6 dice throw) isn't the same as you can't rule out the previous two throws. Even if each throw of the dice is always 1/6 a darter does have the own mind. Confidence and personal experience mixed with nerves then you can't deny the previous two throws. Independence and probability in mathematics or dice throw is one thing but the mind of a darter is another.
Darts and they're relationship with the thrower can't be denied and even the best players can't completely empty they're mind to put the each dart is independent theory into full practice.
But this is only my opinion

Yes, it's all in the mind. Nothing else. That's why it is so important to get it out of your head.

The player makes the 170 so hard to hit, because he just thinks hitting the last dart is so extremely rare. We've all been there, your heart goes like crazy after hitting the first two darts.

Robots are a great comparison. They just wouldn't feel different, and throw just to the probabilities. That pretty much gives evidence on the mathematical point.

What makes it easier to get calm your nerves down than understanding your chances? Some players might pull of great thing with the biggest adrenaline rush, but truth is - most will fail. You mind goes boom and your hand starts shaking like crazy. Take it as it is - a stochastic game.

It may be something completely different talking about a stochastic model than your mind games. But by all means, don't make the mathematical easier finish the harder one in your mind! Don't think "I just hit a treble now I will surely fail with the next dart." There is absolutely no reason for that.

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Obviously this is an offshoot of Cyanide's thread and the debate continues here on ico's thread.

This is all about one's personal experience in which case all those who got involved in presenting their views based on their individual experience - are correct.

I would like to declare or propose a deadlock on this type of discussion.

As if one imposes his belief over the other who refuses because he has his own, it will only end to nothing.

Please let's call it a draw or ceasefire. . . otherwise. . .I'll start a barrage of Yo Momma jokes again.

I am a big fan of true gentlemen dart players like you guys so please allow me to continue to respect all of you.

Thanks a lot!
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it'll be a pretty quite forum if we stop discussing things people want to talk about . Cys thread was great this also is a good thread offering a different view . who is right ? i don't care ,but it'll all food for thought
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Ok, I'll sum up Cyanide's joke again:

With two darts in your hand, it is easier to take out 110 than it is to take out 72.

If this is what people want to believe (hitting T20 D25 is easier than hitting T20 D6) - so be it. I hope to meet many of you at the board.

Enough said. No Yo Momma jokes for me - this claim was funny enough.

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Thanks for doing this ico.
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This is a fair discussion point I agree, but cy's thread got a bit heated. We can discuss anything to great length and detail as long as its kept in good spirit.
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