Biased Dice and the Higgs Boson

Suppose for a moment that you own a pair of dice and you're convinced that one of them is prone to roll one number slightly more often than the other numbers. You don't know which of the two dice is the biased one.

How many rolls would it take to prove beyond five sigmas that there is in fact a bias of some kind in one of the die and how could you discern for which face?

I threw the Higgs Boson in the title because I think this is the type of task that the people at the LHC have to deal with to "prove" that the Higgs Boson exists.

The Atlas experiment team has upped its level of certainty for Higgs-ness in a paper [PDF] for Physics Letters B, putting the sigma level at 5.9, which translates into a one-in-300-million chance that the observed elementary particle isn't the highly sought-after mass-giving boson.

Quote: s2dbaker

How many rolls would it take to prove beyond five sigmas that there is in fact a bias of some kind in one of the die and how could you discern for which face?

The answer to that question is not as clear as you might think. If you keep rolling until the results are five standard deviations off from expectations, at some point you will pass the test for bias. This is why the experiment should be stated from the beginning. Personally, I would toss the dice 1,000 times and do a chi-squared test on each one at the end.

Quote: s2dbaker

I threw the Higgs Boson in the title because I think this is the type of task that the people at the LHC have to deal with to "prove" that the Higgs Boson exists.

In all seriousness, I doubt smashing dice together at near light-speed would accomplish much beyond generating a huge electrical bill to pay.

Would it help to think of the "dice" as Protons and the faces are the energies they are allowed to give off?

Splendid idea !

Quote: s2dbaker

Would it help to think of the "dice" as Protons and the faces are the energies they are allowed to give off?

Not really.

But we could test whether or not the dice are entangled. Take one die off a pair and move halfay around the world. Toss it, and if the die that stayed home changes state at once then they are entangled.

Quote: Nareed

But we could test whether or not the dice are entangled. Take one die off a pair and move halfay around the world. Toss it, and if the die that stayed home changes state at once then they are entangled.

I don't think that's the way dices (or other objects like quantum particles) are tested for entanglement. If you would be observing a "instant change in state due to entanglement with another object" you will get the Nobel price for sure.

If you (A) would want to test your dices for entanglement: After moving one of the dices half around the world, and agree with the other scientist (B) on throwing the dice regulary (say every minute). Each scientist (A and B) records his throws and repeats it until this projects runs out of funding.
In the last week each scientist examines his own records, and comes to the conclusion that his dice is fair (*), i.e. he verifies that - with respect to the finite number of his throws - each side "1" to "6" has the same probability (1/6).


Now to test for *entanglement* of the dices, both scientists meet somewhere and combine theirs records. Since they agreed on throwing the dices every minute, for each minute they can make pairs "A-B" for the dices. In the same way they examine the probability of each of the 36 "dual-throws" 1-1, 1-2, ... 5-6, 6-6 and test whether theirs occurences are consistend with the probability of 1/36.

If they find a signifiant deviation from 1/36 (although each dice have been estimated to be fair by itself), they call theses two dices - at least during the course of the experiment - entangled.



(*) even if each dice is not fair, you can still test for entanglement. But for simplicity lets assume they are.

Quote: Nareed

Not really.

But we could test whether or not the dice are entangled. Take one die off a pair and move halfay around the world. Toss it, and if the die that stayed home changes state at once then they are entangled.

Think of all the Hard-Ways you would get. Dice entanglement .. I'll work on that.

Quote: MangoJ

I don't think that's the way dices (or other objects like quantum particles) are tested for entanglement.

Oh, well, you probably have to shoot them with laser beams or something first...

Quote:

If you would be observing a "instant change in state due to entanglement with another object" you will get the Nobel price for sure.

Well, an electron cannot be observed closely enough or fast enough to see this take place in an instant. For that matter no one's seen or even detected a Higgs' boson, either. But something as big and as slow as a die can be measured in different ways.

But then if you knew the state fo the die you wouldn't know where it was, would you?

What I really want to do is find the Shackleford boson. Theory states it can imbues all sorts of actual and virtual gamblions with variance ;)

Quote: s2dbaker

Think of all the Hard-Ways you would get. Dice entanglement .. I'll work on that.

A good start would be to carve pips on the surface of an electron :P

Can Dark Side Shooters use anti-protons?