Virtually Certain - some maths

[Ian_S]

Firstly, apologies if this a bit of a geeky post, it's something I was thinking about today and just fancied sharing my thoughts.

I was thinking back to a round the other week when a tee shot of mine was heading towards some water. The water is actually a stream in a little valley, so you lose sight of the ball as it goes into the valley but there is enough grass there that if you're lucky it stops short of the stream. Anyway, the grass was a little longer so there was a chance the ball could have been caught in it but I felt it was probably in the water. We get to the spot, have a look around and can't see my ball anywhere, in the water or out. For me, I'm virtually certain that it's gone in the water, especially given that we can't see it on the slope but a FC says "but we didn't see it go in".

My thought today was "can I put some estimates on this and get to a number, and decide whether that counts as virtually certain?"

Here's where the maths comes in. If I say my initial estimate is that there's a 50:50 chance it's in the water, if it's on the slope I'd find it 95% of the time and if it's in the water then I'd find it 10% of the time, then I can use Bayes' theorem to update that initial estimate. Plugging those numbers in, that fact that we can't find the ball changes the probability that it's in the water from 50% to 95%. But is 95% 'virtually certain'?

Suppose I have two coins, one doubled-head and one not. I pick one at random, toss it and tell you it's a head. I'm sure no-one would say for any certainty that I picked the double-headed coin from that one toss. But how many heads in a row would it take for you to be virtually certain that I've got the double-headed coin? Four, six, a dozen?

4 heads in a row gives a 94% chance that I've picked the double-headed coin, 6 is 98% chance and 12 is more than 4000-to-1 against that I've got the fair coin.

So I wonder, what do you class as 'virtually certain'?

FWIW, if the grass is longer in my example and the chances of finding it given that it's in the grass is only 20%, then the probability that it's in the water is 53% - definitely not virtually certain.

 

[Ethan]

Virtually certain is akin to the legal concept of beyond reasonable doubt. If there is a large water hazard surrounded by a thick collar or rough and the ball probably bounced before heading for the hazard, then you can't be virtually certain the ball is in the water.

Your theory basically goes that you invariably find it if it isn't in the water, but rarely do so if it is, therefore if you can't find it it must be in the water.

How do you know that there are not examples which you mistakenly assume are in the water, and thus not part of the 95%, where the ball was lost another way not in water, down a rabbit hole, stuck in a small bush, lifted by overflying vultures etc etc?

As far as I know, Bayes didn't play golf, otherwise his concepts of prior and posterior probabilities would have had more uncertainty built in.

 

[DaveM]

Someone's having a quiet day!

 

[Foxholer]

Given that Probability is all about not being certain, is it really applicable to this situation?

And, for Normal Distribution 94% might be, just, greater than 'Very Probably' but is not as high as 'Almost Certain' (99.7%)!

Interesting application of Bayes though!

 

[Colin L]

Good grief! Words are my specialism - can't cope with this kind of Maths. (@Atticus: our Maths is your Math)

Dear Foxholer, whatever it is you are on, I don't want any of it

Keep it simple: is there any possibility that your ball could be lost in the surrounding grass? If yes, then you don't have virtually certainty.

And I am 99.9% virtually certain that I'm right.

 

[Foxholer]

Good grief! Words are my specialism - can't cope with this kind of Maths. (@Atticus: our Maths is your Math)

Dear Foxholer, whatever it is you are on, I don't want any of it
....

:rofl:
Stats and Computing (and a bit of Physics and Astronomy to add to the geekiness) got me through University - in the very early 70s!

 

[Ian_S]

How do you know that there are not examples which you mistakenly assume are in the water, and thus not part of the 95%, where the ball was lost another way not in water, down a rabbit hole, stuck in a small bush, lifted by overflying vultures etc etc?

On the contrary, if the 95% estimate is accurate then 1 time in 20 the ball will be lost and not in the water. But that case is allowed for in the definition of 'virtually certain'. It is permissible to be both virtually certain and wrong (in much the same way as we've all missed virtually certain one foot putts).

 

[williamalex1]

The ball can be lost within limits the water hazard, without it actually being in the water. yes ?

 

[Colin L]

The ball can be lost within limits the water hazard, without it actually being in the water. yes ?

Yes it can. Apart from there being dry land within a water hazard that does have water in it, a water hazard does not necessarily have water in it all the time.

 

[atticusfinch]

These standards intentionally omit any kind of quantitative measure because they depend on the facts of the circumstance which vary in every case.

 

[Ian_S]

These standards intentionally omit any kind of quantitative measure because they depend on the facts of the circumstance which vary in every case.

Indeed, which causes no end of trouble. I know they replaced 'reasonable evidence' with 'known or virtually certain' to try to make things better, but that still doesn't resolve the problem.

'Known' is fine and is a question of fact. 'Virtually certain', however, is a matter of opinion - although the rules try to pretend it's a question of fact. I may see a ball go down a slope towards a hazard and, upon reaching the hazard and not seeing the ball, judge that it's very unlikely that the ball has disappeared anywhere other than into the hazard. However a FC, presented with the same facts (slope, ball flight, conditions, etc), may judge that there is reasonable doubt and that there's a slim but not-negligible chance that the ball isn't in the water hazard. It's not their job to then find your ball so if they say "I'm not virtually certain that it's in there", if your ball is in the hazard then you could well be screwed. It all depends on whose opinion matters in these cases.

It is much the same as my double-headed coin example before. If I ask different people to jump in once they are 'virtually certain' that I'm using the rigged coin, then I will get a wide range of what counts as virtual certainty.

 

[atticusfinch]

In a group of average people you will get average responses about the meaning. But golf rules questions in a tournament are answered by experienced trained people. Their opinions of VC will be more uniform and closer to what is intended.

 

[drdel]

Golf shots are not mutually exclusive events because your skill and capabilities are the common factor and you may have played the hole frequently before and thus with your high level of skill your shots will have a very tight dispersion :clap:

Bayes theory doesn't then apply.

If you knew from previous shots at this hole that your average shot length was either long enough to reach the stream or only long enough to reach the grass bank then you have an average expectation.

If you then know that your carry/length of shot varies by say 2, 5 or 10 yards then you have factual data set that allows you to say with a reasonable probability (68%, 1 deviation) or be 'virtually certain' (95%, 2 deviations) whether your average shot would have taken you into the stream or short of the water.

 

[Colin L]

Golf shots are not mutually exclusive events because your skill and capabilities are the common factor and you may have played the hole frequently before and thus with your high level of skill your shots will have a very tight dispersion :clap:

Bayes theory doesn't then apply.

If you knew from previous shots at this hole that your average shot length was either long enough to reach the stream or only long enough to reach the grass bank then you have an average expectation.

If you then know that your carry/length of shot varies by say 2, 5 or 10 yards then you have factual data set that allows you to say with a reasonable probability (68%, 1 deviation) or be 'virtually certain' (95%, 2 deviations) whether your average shot would have taken you into the stream or short of the water.

I couldn't have put it better myself.

 

[duncan mackie]

.... there is enough grass there that if you're lucky it stops short of the stream. Anyway, the grass was a little longer so there was a chance the ball could have been caught in it....

for me this suggests that there is sufficient possibility the balls in the grass not the water for K or VC that it's in the water not to exist - however you have to be there to make such calls.

I use a loose rule of thumb that if a player spends more than a minute looking outside the water hazard for their ball he's not sufficiently committed to any certainty it's in the water!

 

[Foxholer]

If you then know that your carry/length of shot varies by say 2, 5 or 10 yards then you have factual data set that allows you to say with a reasonable probability (68%, 1 deviation) or be 'virtually certain' (95%, 2 deviations) whether your average shot would have taken you into the stream or short of the water.

Except 2 Standard Deviations/95% is only 'very probably'! Almost certain (99.7%) (and maybe sufficient for 'virtually certain') would be 3SDs! Bayes Theory/Rule/Law would still apply - the numbers (probabilities) would simply be different.

@ColinL What did you say the Gaelic/Scots was for Cow Poo again? :rofl:

 

[williamalex1]

Except 2 Standard Deviations/95% is only 'very probably'! Almost certain (99.7%) (and maybe sufficient for 'virtually certain') would be 3SDs! Bayes Theory/Rule/Law would still apply - the numbers (probabilities) would simply be different.

@ColinL What did you say the Gaelic/Scots was for Cow Poo again? :rofl:

Foxholer's posts maybe :rofl:

 

[Colin L]

Except 2 Standard Deviations/95% is only 'very probably'! Almost certain (99.7%) (and maybe sufficient for 'virtually certain') would be 3SDs! Bayes Theory/Rule/Law would still apply - the numbers (probabilities) would simply be different.

@ColinL What did you say the Gaelic/Scots was for Cow Poo again? :rofl:

Shairn or sharn.

 

[Foxholer]

Shairn or sharn.

That's the stuff - entirely applicable to most statistics.

And certainly to my posts when viewed through a Brandy or 2 by the 'man' from Uddingston!

 

[williamalex1]

That's the stuff - entirely applicable to most statistics.

And certainly to my posts when viewed through a Brandy or 2 by the 'man' from Uddingston!

Your correct Foxy , the brandy thinks , maybe you should change your signature from Fox to Ass suits you better .:rofl: