Calculating Statistical Likelihood Knowing Standard Deviations

[TomVassos]

TL;DR

Can you calculate the likelihood that something has already occurred, if you only know, with 95% certainty, when it is likely to occur in the future?

Let’s say that we know, with 95% confidence, that something is likely to occur when the Universe is between 10⁵⁸ and 10⁵⁴⁹ years old.

What is the statistical likelihood that it has already occurred in the first 13.8 billion years of the Universe’s existence? (1.38 X 10¹⁰ years)

I know the answer to this question is going to be a miniscule number, very close to zero, but I would be eternally grateful if anyone on this forum could answer this question.

Thanks in advance for your help!

Tom

 

[Dale]

From only what you have specified here, there is no answer. We would additionally need to know how the likelihood of the event changes over time.

The type of statistics you are looking at is most commonly used to predict the failure of a part. If a part is likely to fail due to manufacturing defects then the likelihood of failure will be very high in the beginning, and lower later. If a part usually fails due to wear and tear then the likelihood of failure will be low in the beginning and increase as time goes on. If failure is just random then the likelihood would be constant.

That last case is the easiest.

 

[TomVassos]

But by knowing with 95% confidence when this event is likely to occur, doesn't that tell us the exact shape of the Bell curve that describes its probability? Can't that be used to determine the probability of that tiny little slice at the far left side of the Bell curve?

Tom

 

[PeroK]

I know the answer to this question is going to be a miniscule number, very close to zero,

https://www.merriam-webster.com/dictionary/minuscule

 

[Dale]

But by knowing with 95% confidence when this event is likely to occur, doesn't that tell us the exact shape of the Bell curve that describes its probability?

No. With no other information, all we can say is that the probability is less than 5%. It could be minuscule or it could be close to the 5% limit.

We don’t know that it follows a bell curve. In fact, we actually are certain that it does not follow a bell curve. A bell curve would include times before the beginning of the universe. Often that is such a minuscule probability that it doesn’t matter, but here when you are calculating something also very small it could matter.

 

[TomVassos]

Hi Dale, thanks for your questions. To give you a bit of background information, this is the likelihood of something called "vacuum decay" occurring. (It is when a Higgs Boson decays from a false vacuum to a true vacuum state, causing something called a bubble nucleation which will begin to destroy the universe at light-speed.)

I have seen the Harvard research results that do in fact indicate that it follows a Bell curve, so please use that assumption. Also, in our Universe, another assumption we can make is that vacuum decay did not occur prior to the Big Bang.

So, if we make those two assumptions, can we now make this calculation?

Thanks!

Tom

 

[PeroK]

I think there's a 95% probability you've already asked this question!

https://www.physicsforums.com/threa...-using-standard-deviation-statistics.1059196/

 

[TomVassos]

Yes, I was asked to remove any reference to any paradoxes, etc., which I have done so now I have just posed it as a straight math/statistics question. Hoping someone on PhysicsForums has the smarts to solve this math dilemma for me... :)

Tom, Canada

 

[Dale]

I have seen the Harvard research results that do in fact indicate that it follows a Bell curve, so please use that assumption. Also, in our Universe, another assumption we can make is that vacuum decay did not occur prior to the Big Bang.

Those two assumptions are incompatible with each other.

 

[TomVassos]

Hi Dale, not sure why you say that. Picture a Bell curve where the far left side of the curve begins at time zero, when the Big Bang occurs. And it ends at the far right side, somewhere in the range of 10^607 years (549 + 58). The area of the Bell curve prior to 10^58 years should be the same as the area beyond 10^549 years.

 

[TomVassos]

Even if you disagree with the assumption that time zero started at the Big Bang, can we still make this calculation?

 

[Dale]

Picture a Bell curve where the far left side of the curve begins at time zero,

Bell curves go from -infinity to +infinity. There is no such thing as a bell curve that starts at some value.

 

[Dale]

Here is my take on this as a statistics exercise. As stated the problem is impossible. You could make it possible with any one of many sets of assumptions.

However, the problem is about the probability of the extreme tails of the distribution. Such probabilities are inherently unreliable and sensitive to even small violations of the assumptions. So basically the answer you get tells you about your assumptions rather than the physics.

 

[TomVassos]

Hi Dale, let's take your example about the likelihood of a particular part failing. If I install the part on January 1, looking at the bell curve, can I not determine the likelihood that the part will fail sometime between January 1 and January 15th for example? If the part is installed on January 1, by definition, the Bell Curve must be at zero on January 1 since it is impossible for it to fail before it is installed. Am I missing something?

 

[Dale]

let's take your example about the likelihood of a particular part failing. If I install the part on January 1, looking at the bell curve

It’s not a bell curve. Usually for part failure we will use an exponential distribution or a Weibull distribution. An exponential distribution would model a part that would be equally likely to fail at any time. A Weibull distribution could be used to model parts that mostly fail by wearing out or mostly fail due to manufacturing defects. The exponential distribution is the easiest.

can I not determine the likelihood that the part will fail sometime between January 1 and January 15th for example?

You can do that for an exponential or a Weibull distribution. There are also many other distributions you could use.

If the part is installed on January 1, by definition, the Bell Curve must be at zero on January 1 since it is impossible for it to fail before it is installed. Am I missing something?

The bell curve, by definition, goes from -infinity to +infinity. That is why it is not used to model part failures. See https://en.m.wikipedia.org/wiki/Normal_distribution where it says that the support is all real numbers.

 

[TomVassos]

Dale, I see what you are saying. Thanks so much for taking all that time to explain everything.

I took a closer look at the research about when vacuum decay is likely to occur, and Andraesson et al. state that it will occur between 10^88 and 10^291 years with a 68% confidence level, and between 10^58 and 10^549 with a 95% confidence level.

Their research paper is located here:
https://journals.aps.org/prd/pdf/10.1103/PhysRevD.97.056006

But in reviewing this paper, it looks like they have calculated this probability right in the research paper. They seem to state that the probability that vacuum decay has already occurred is 10^-516.

That’s pretty close to zero! :)

Tom

 

[Agent Smith]

B is for basic.

If by "confidence" you mean probability then ...
there's a 5% chance that the event has already occurred ($< 10^{58}$) or will occur outside the given range ($> 10^{549}$). How would you distribute the 5% probability between these two (past and future) regions? Is there a logic to the distribution or is it anything goes? Remembering that $>10^{549}$ years is $+\infty$, the portion of the remaining 5% probability set aside for $< 10^{58}$ years should be really small (as you correctly stated), but is it $0$?

 

[Dale]

There are a few different intervals that are used. But by far the most common is the equal tailed interval. For a 95% interval that would have 2.5% on the left and 2.5% on the right. If it is not explicit, then this is probably what they mean.

There are also one tailed intervals, which have 5% on one side and none on the other.

There are also highest density intervals which go somewhere in between so that you get the smallest possible total width to capture 95% without regard to how much is on one side or the other.