Alot of prolific mathematicians can win hugely prestigious awards for only a fraction of their total work .

Quite high

if someones dad dies of cancer what are the odds (more than a person whos father died from long age) of that guy diying from cancer, too?

How did I miss your reply?!?!

heh cool link with the air molecules. I've been thinking that the threshold to say it is impossible is entirely dependent on the context that it is in, some situations it would be perhaps 10^-10, others perhaps .05. So when I'm driving in my car and pondering the risks, I can deem that it's impossible for the driver who is traveling in the same direction and parallel/adjacent to me to suddenly swerve and crash into me, but perhaps a road safety scientist guy wouldn't omit this

Answering this question requires a model and data, neither of which I have.

Inspired by a conversation via private messages, I wanted to post the following exercise. It requires basic knowledge of measure-theoretic conditional expectation and conditional probability.

In modal logic, propositions are not only true or false, but can also be classified as possible or necessary. An obvious intuitive connection to probability suggests that an event A is possible if P(A) > 0, and necessary if P(A) = 1. To accommodate different kinds of possibility and necessity, we might say that A is G-possible if P(A | G) > 0, and G-necessary if P(A | G) = 1.

The so-called Axiom S5 of modal logic is considered by some to be unintuitive and controversial. It states that if something is possibly necessary, then it is in fact necessary. Prove the following probabilistic analogue:

If P(P(A | G) = 1 | G) > 0, then P(A | G) = 1.

More specifically, let (Ω, F, P) be a probability space, G a sub-sigma-algebra of F, and A an event in F. Prove that for P-a.e. w in Ω, the following are equivalent:

1. P(P(A | G) = 1 | G)(w) > 0,
2. P(P(A | G) = 1 | G)(w) = 1,
3. P(A | G)(w) = 1.

In modal logic, the analogous fact is that possible necessity, necessary necessity, and necessity are all the same thing.

Also prove that for P-a.e. w in Ω, the following are equivalent:

1. P(P(A | G) > 0 | G)(w) > 0,
2. P(P(A | G) > 0 | G)(w) = 1,
3. P(A | G)(w) > 0.

In modal logic, the analogous fact is that possible possibility, necessary possibility, and possibility are all the same thing.

My understanding is that the interpretation of possibility as P(A) > 0 only holds for physical possibility. A probability space (as I understand it) would cover anything that is physically possible. Thus, if we use "possibility" to refer to metaphysical or logical possibility rather than physical possibility, then we cannot interpret it as a probability.

For example, it is metaphysically possible, but physically impossible for a neutrino to travel faster than the speed of light. Thus, the P(neutrino traveling faster than c) = 0. But this is metaphysically possible. Logical possibility is broader yet. So, for instance, we say that it is logically possible for the ball to be completely red and completely yellow (i.e. it is not logically contradictory), but that this is neither physically nor metaphysically possible.

Is this wrong? Can probability range over more than physical possibility?

A probability space is an abstract mathematical object with no intrinsic connection to anything physical. A connection to the physical is made when the probability space is used as a model and subsequently interpreted. Perhaps there are certain philosophical interpretations of probability under which probability can only model physical possibility. But in my view, such an interpretation would be unnecessarily restrictive.

For example, suppose I have a finite Boolean algebra F of propositions, partially ordered by logical implication and constructed so that p = q if and only if p and q are logically equivalent. We may identify F with the Boolean algebra of subsets of a set Ω in such a way that p is identified with A_p ⊂ Ω, and A_p ⊂ A_q if and only if p logically implies q.

In this case, there are many probability measures P that can be put on the measurable space (Ω, 2^Ω) so that 0 < P(A) < 1 whenever A is a proper nonempty subset of Ω. If we choose one of these P's, then we will have P(A_p) > 0 if and only if p is logically possible, and P(A_p) = 1 if and only if p is logically necessary.

In this example, we used unconditional probabilities, which is equivalent to conditioning on the trivial sigma-algebra. By enlarging the sigma-algebra, we can create notions of possibility that are narrower than logical possibility. However, if there is a notion of possibility which is even broader than logical possibility, then I do not believe probability theory can express it.

I'm really interested in probability theory. And even though I've looked up the Bayesian probability in wikipedia. I feel like it's 1/100 of a large puzzle.

What kind of books would you recommend to anyone interested in probability theory?

I usually get the hang of new things/concepts quite fast imo & it doesn't have to be a probability for dummies book.

are you familiar with Taleb and Mandelbrot's work? If so do you have any opinions or comments?

Also, can you name some places you find stats and probs significantly abused? Does research in medicine and economics seem to be sound to you?

An issue i've found is statistics are well applied to data in many fields but there is grave errors in the data collection and aggregation process. Are there methods to account for this? Is there a margin of error for the margin of error? Is there some rigor to ensuring correct interpretation and consistency along the process of collection to calculation?

(i know these are not well devised questions, i apologize. Please do your best if you have the time)

It really depends on your background. A nice moderate-level book is Probability Essentials by Jacod and Protter. If you need something lighter, you could try to read Ross.

I know Taleb writes popular books, and Mandelbrot named fractional Brownian motion. That is about all I know.

I think many people, including some professionals in science, medicine, law, philosophy, and so on, are unaware of the depth and richness of mathematical probability theory, and consequently believe that probability is a high-school or undergraduate level subject on par with basic algebra. As a result, many people feel that their basic statistics course, possibly taught by a psychologist for example, gives them a proficiency, or an even an expertise, in the subject. In my view, this is what leads to abuse and unfortunately, I believe this situation is quite common.

I am not qualified to say, but I would guess that, like most disciplines, some research is sound and some is not.

These questions are a bit too vague for me to know how to answer. Depending on what you mean, your questions may be better directed to a statistician. But I can say that there is rigor in the mathematics of probability theory, and probability is not statistics. Within statistics, Bayesian methods, in principle, use only the mathematics of probability theory and are required to make their assumptions explicit. Non-Bayesian methods may use algorithms which are not contained in the pure theory of probability, and these algorithms may have implicit assumptions about the nature of the data and how it was collected.

I was one of those teachers and found that it was impossible to do more than cover the basics of why a few statistical tests work and which one would be most appropriate for a given type of data. Even that was a struggle in a semester even though a prerequisite for the course was a prior prob/stats course.

And, as you have mentioned, statistics is not probability.

***

In case you haven't already covered this, what are the career choices for someone with a strong background in probability?

Careers involving probability and statistics

On a more personal level (always a better and more interesting story), what you gonna do?

Do you think that stochastic or complex systems (formerly chaos theory) has a better chance of decent description of financial systems? (understood if you aren't an expert on the specific field, but would enjoy your opinion and reasons for it.)

Gonna do? He a mother-****in' hustler---you betta ask somebody!!

Hi, sorry for the simple question.

Playing 10-player sit 'n goes in a set of 3, all starting roughly at the same time, I figured that the average player (or a player who finishes in the money 30% of the time) will finish out of the money in all three games 34.3% of the time (.7^3) and therefore in the money in one or more games within the set 65.7% of the time. They will also finish all 3 games ITM 2.7% of the time. What is the percentage chance that they will cash in 1 of 3 and in 2 of 3 in the set?

I am a professor of mathematics. I expect to continue with this career.

Chaos theory says that we cannot predict the future because of sensitivity to initial conditions. Probability theory can say the same thing, but then we assign probabilities and we keep going with our analysis. So whether it is financial systems or something else, probability theory has more potential than chaos theory because it has all the same tools and then some.

Also, thank you for the opportunity to take advantage of #7 on the list of the Top 10 reasons to be a probabilist.

-grunch
Why do you call yourself a probabilist? What's the difference between you and me (mathematical statistician)?

Emphasis on teaching?

I'm partial to geometric and fractal brownian motion models.

Your welcome

In all probability it is better to post this question in the probability forum:

http://forumserver.twoplustwo.com/25/probability/


-Zeno

I hope that you did more than just estimate the probabilities in this otherwise excellent thread.

I doubt that "questions" even approximates a gaussian distribution, and it is somewhat questionable whether "forums" is nominal.

I have a question:
If you were to choose what kind of work you do in your career as probabilist what would it be ?
Would it be theoretical work on chosen area ?
Would it be some work on real life problems ?
Some mix of those ?
Something else ?

How does it relate to what you actually do ?

Can you really call yourself a mathematical statistician if you don't know the answer to this question?

A probabilist posts preprints here. A mathematical statistician posts preprints here. If you look at some of the posted preprints in those two locations, you can get a better understanding of the difference between the two fields.