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.

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.