# B What is the least likely event that has been observed?

1. Jan 19, 2017

### Grinkle

I don't know if this is the best forum for my question, hopefully this is not just a lead balloon question.

I expect that the statistically least likely events to ever be observed have been observed in a particle accelerator. Does anyone have any idea what takes the award for the statistically least likely event to ever be seen and to have been seen goes to? Particle accelerators are engineered to make very unlikely events much more likely, so maybe I am just asking what the effectiveness of the accelerator is and how lucky we have been with that effectiveness.

I labelled the question B, I am a bit self-conscious that it may be a nonsense question.

2. Jan 19, 2017

### Orodruin

Staff Emeritus
This question is not really well defined. In general, for any observation that draws from a continuous probability distribution, the probability of any given outcome is exactly zero yet the observation must give a result.

This is not really the case. Particle accelerators do not make events more likely, they simply collide a large number of particles such that a few out of the large number of collisions give results that would be unlikely as a result of any given collision.

3. Jan 19, 2017

### Staff: Mentor

How do you define the likelihood?

To keep the particle accelerator example: Take a random lead-lead collisions with 10000 hadrons produced that fly through the detectors. The probability to have exactly those 10000 particles with their momenta? Utterly negligible. The probability to have about 10000 hadrons with a similar energy and particle type distributions? Close to 1, happens thousands of times per second.

In terms of interesting events:
The LHC experiments had about 1016 proton-proton collisions so far. Pick the event with the highest-energetic electron/positron pair: You know have something that happened just once in 1016 collisions. Pick the event with the highest-energetic muon: You know have something that happened just once in 1016 collisions. And so on. You can also look at the expectation, that won't be so much different from 1016.

For the neutrino detector OPERA, CERN shot about 2*1020 protons on a target, producing many muon and electron neutrinos. 5 of them oscillated to tau neutrinos and got detected at OPERA: Events that happen only once every 4*1019 protons shot on a target. Are those events less likely? Well, most neutrinos missed the detector - do you want to include those in the calculation? Why do we count protons instead of neutrinos produced? Should we count the tau neutrinos, and take only the probability that they get detected? Or only if they fly through the detectors?

4. Jan 19, 2017

### Grinkle

Thanks, both - I realize its a sketchy question.

I'll try to clarify -

In a different thread, part of the discussion often came back to whether there is observation of extremely unlikely events, and whether it matters if there is observation or not of extremely unlikely events when discussing their inevitability given enough trials, and that made me curious about whether we can put some current bound on the experimental confirmation of long-shot statistical predictions having been observed.

As a practical matter, one cannot flip a single coin enough times to experimentally prove that eventually one gets 'x' heads in a row, if one picks a large enough 'x'. It becomes impossible to carry out the experiment even in a theoretical sense, since it quickly will be predicted to take longer than the sun will be burning even for a not very large 'x'. But perhaps there is a direct mapping of a 50/50 event in a particle accelerator that puts enough 'coin tosses' in parallel that for a coin toss, one can claim experimental confirmation that indeed one does observe the exact statistical equivalent of 'x' heads in a row given enough attempts. If so, it would be interesting to me to know what the largest observed equivalent 'x' is.

It may seem foolish of me to look for experimental confirmation of the predictions of something as basic and intuitively correct as the statistics of a coin toss - I do not doubt the predictions these statistics make, and I don't expect that statistics breaks at some very small but non-zero probability level. I am just wondering what it is possible to claim experimental confirmation of.

5. Jan 19, 2017

### stoomart

The 50/50 mapping you are talking about could be the median electron/positron pair energy from the examples @mfb gave above. I've looked into getting LHC event metadata to perform this kind of analysis before, but unfortunately it's not publicly available for download.

6. Jan 19, 2017

### Staff: Mentor

You cannot find a median because the low-energetic part is not detected. The electron/positron invariant mass spectra can be obtained from released datasets (7 TeV data from 2011). Those spectra are published as well.
Such a mapping is not well-defined. Coin tosses give well-defined discrete outcomes, particle collisions do not.

7. Jan 19, 2017

### OCR

True ! ...

Lol...

8. Jan 19, 2017

Staff Emeritus
I don't think this is well defined. Events are like snowflakes - every one is different.

Which is less likely in poker? A royal flush? Or a 2 of spades, a 4 of clubs, a 7 of diamonds, an 8 of diamonds and a 9 of clubs?

9. Jan 19, 2017

### stoomart

Lol...neither, only the latter gets filtered out like the uninteresting events in LHC.

10. Jan 20, 2017

### Grinkle

I think -

Given a process that I choose to model as 0.5/0.5 probability for two outcomes and 0.0 for any other outcome (I choose to model a coin toss that way) that I can predict how many runs of the process it will take before I am 99.999 (or however many 9's I care to have) likely to have seen at least one sequence of outcomes that is what I define.

I am quite sure there is no reason in the statistics or the predictive exercise that 100 heads in a row needs to have any physical significance.

I think I am applying statistics properly in posing the question, but I've been very wrong about things I've been surer of, so correct me if I'm off.

Watching a random process and attempting to predict when a specific physically uninteresting sequence will occur is arguably an experiment of dubious value - if that is your point, I get it.

11. Jan 20, 2017

### Staff: Mentor

The coin tosses are fine. They just don't have a proper equivalent in particle physics.

12. Jan 20, 2017

Staff Emeritus
Grinkle, I have no idea how what you wrote follows from what I wrote.

13. Jan 20, 2017

### Grinkle

Then I accept that it doesn't, I must have misunderstood.

I read you to be saying that assigning probabilities for different system states with no thermodynamic distinction between them is not a well defined exercise, and I wanted to see if I am going off track somewhere.

14. Jan 20, 2017

### PeroK

A Royal Flush is more likely as there are four of those! Which sort of emphasises your point!

15. Jan 20, 2017

Staff Emeritus
Exactly.

16. Jan 22, 2017

### Meir Achuz

human existence

17. Jan 22, 2017

### e.bar.goum

I would have said that the question of the OP is perfectly well defined in the context of cross-sections: "What is the process with the smallest cross-section ever to have been observed?".

18. Jan 23, 2017

### Staff: Mentor

See the heavy ion collision: The result would be extremely small (<10-100 fb, probably hundreds of orders of magnitude smaller) if you consider all particles in a random boring event within the measurement uncertainty, and it approaches zero if you don't take the measurement uncertainty into account. What does that tell us?