Definition of probability.

[neutrino86]

During my schooling, i read from an elementary probability book that the probability of an event occuring in a sample space( i donot know if it was a countable set or whatever else is likely), is a number obtained by letting the no of trials tend to infinity and then drawing up a number of success/failure figure out of it, in the end obtaining the probability structure of an event.

 

Well, now i read from apostol that during the course of formation of games-of-chance theory, and thereafter probabilty/measure theory, many mathematicians had suffered in coming up with a convincing(?) definition of probabilty.Now come to think of it I feel disturbingly uncomfortable with the definitions I have learnt so far, including the one Mr. Apostol makes in in his book Calculus as:-

A set function P defined on a boolean algebra whose elements are subsets of a given finite set S, is said to be a probability measure if

1.P is finitely additive

2.P(X) >0 for all X in S.

3.P(S)=1.

(and that to it pertains to finite sample spaces. let us just stick to that as much as possible.)

 

All true no doubt, but I wish for some assistance in getting comfortable with the definition, and perhaps a few examples highlighting its possible symmetry with the previous definition of probabilty.

 

And ofcourse any other notions of probability that exist are welcome, as I perhaps may need as many "comprehension crutches" as possible.

 

 

Thanking you in advance.

[matt grime]

Apostol's is the correct formal definitoin of a probability space (the possible outcomes) and measure (the probability of some event) and is an axiomatization of the informal idea you're used to. The strong law of large numbers means that the two ideas are equivalent.

 

Take the THEORETICAL ideal of rolling a perfectly fair die once. The sample space is {1,2,3,4,5,6} and we assign the measure of P(die reads x)=1/6 for all x in the set {1,2,3,4,5,6} I don't actually need a fair die to hand to talk about this. This is the point of abstract mathematics: we make abstract models of things in real life then we can forget real life entirely and think purely of the model.

 

If you simulate the sampling of a large number of rolls then the proportion of times x occurs will approach 1/6.

 

So we idealize and talk about theoretical random variables. There is no such thing as a perfectly fair dice for instance.

[neutrino86]

THankyou sir...That very much clears off the fog.

[softdragonz]

 

So we idealize and talk about theoretical random variables. .

 

 

Random variables have a problem....they are random in nature.......even if we take a 100000 attempts in the dice game, it is possible to get the number 6 all the 10000 times......The random nature requires a more precise definition.....Is there any mathematical formula by which we can find out random numbers??? May be probability could be defined through it

[Dave]

That is the entire point of probability. You cannot predict random variables. You can only calculate the possibility of a particular event occurring.

[matt grime]

Random variables have a problem....they are random in nature.......even if we take a 100000 attempts in the dice game, it is possible to get the number 6 all the 10000 times......The random nature requires a more precise definition.....Is there any mathematical formula by which we can find out random numbers??? May be probability could be defined through it

 

I wouldn't know where to begin with the problems in this.... Oh, no, I do: how about: that makes no sense? Really, it doesn't. There are words there and all the words individually make sense, but not together.

[neutrino86]

Random variables have a problem....they are random in nature.......even if we take a 100000 attempts in the dice game, it is possible to get the number 6 all the 10000 times......The random nature requires a more precise definition.....Is there any mathematical formula by which we can find out random numbers??? May be probability could be defined through it

 

 

I donot think that any finite number of trials results(in the case where random variables come in) in the probability of the particular event to occur, rather , it tends to the probability only when the governing equation(s) consisting the number of trials, have that particular variable tend to infinity...ie one can never experimentally observe a tendency of the probability function to tend to a real number.

In some cases, the number of such equations could turn out to be terrifyingly large, and hence another definition of probability would be required.

[Dave]

I suggest you pick up a coin, toss it, record the result and carry on for about 3 hours. After that, work out the percentages of heads and tails, then check to see whether they give a 50/50 average - or close to, at least. The point is that you could never experimentally "prove" anything in probability, but at least if you perform an experiment a large number of times then you can verify that the functions you come up with are at least close to being right. You certainly do not need "another definition of probability", whatever that might mean.

[the tree]

Probability is the mesurment of how probable an event is. What more do you need?

[matt grime]

Well, that is the nice wooly idea, but it needs formalizing if you are to undertake mathematical analysis of these things as abstract objects. Ultimately you aren't even reasoning about things that 'occur' but just formal random variables. This becomes especially true when one passes to infinite (discrete or continuous) sample spaces.

 

For instance we commonly refer to picking an integer at random, but actually that is a very loose terminology indeed, and needs plenty of clarification as to what you mean if you were to be pressed as to what you actually meant.

[BigMoosie]

For instance we commonly refer to picking an integer at random, but actually that is a very loose terminology indeed, and needs plenty of clarification as to what you mean if you were to be pressed as to what you actually meant.

 

That kind of clarification seems more relevant to a philosopher than to a mathematician.

[shmoe]

That kind of clarification seems more relevant to a philosopher than to a mathematician.

 

On the contrary, it's extremely important for a mathematician. You won't find a mathematician trying to prove results about hand wavy or vague definitions and there are plenty of "the probability a random integer is blah" statements, for example,

 

The probability a random integer is divisible by 3 is 1/3.

 

The probability a random natural number is prime is 0.

 

The probability two randomly selected natural numbers are coprime is 6/pi^2.

 

In these cases "Probability of a randomly selected blah" is in the sense of asymptotic density that can be made precise. What's often overlooked by the casual observer is that there is no way to select a random integer uniformly from the entire set of integers. Since a uniform distro. is what we're after, we end up using uniform distribution on finite sets and taking limits, etc.

[matt grime]

That kind of clarification seems more relevant to a philosopher than to a mathematician.

 

 

If you believe that then perhaps you either did not understand what I meant or maths is not the subject for you.

 

Mathematics is all about definitions. Notice how no mathematician has trouble with infinity yet that doesn't stop people asking dumb questions on it here simply because they do not konw or accept some definitions

[BigMoosie]

If you believe that then perhaps you either did not understand what I meant or maths is not the subject for you.

 

Mathematics is all about definitions. Notice how no mathematician has trouble with infinity yet that doesn't stop people asking dumb questions on it here simply because they do not konw or accept some definitions

 

You're right I didn't quite realise that when meant. I thought you were making an example of picking a random integer from a finite set. Don't get me wrong I was not trying to undermine mathematical definitions in general.

[neutrino86]

yes... i too believe that in places where abstractions are required to simulate real life phenomena, a more formal definition of probability is what is needed...viz in places where an inumerable number of variables set in, none of which can be neglected, for each one of them may have a strong hold on what the next thing that could happen....chaos theory i believe is the best example...& complex models of high atomic mass atoms, etc certainly cannot have a poor mathematician/physicist stuck with infinite trial experiments!

I might not know what a " more formal definition" would look like, but i certainly feel that i could imagine what could be required...

[matt grime]

That isn't a description of chaos theory. Chaotic systems are in some obvious sense highly deterministic systems. The map from [0,1] to [0,1] given by x goes to kx(1-x) [the logistic map, 0<k<=4] is the canonical example of a chaotic system and as you can see there is only one variable, and we can tell its evolution in time explicitly, but the system displays topological transitivity and sensitieve dependence on initial conditions, ie the two properties we (usually) take to mean what the layman calls chaos. I wish I had a pound for every person who's mis-cited chaos theory as an example of something. It seems that the non-mathematical world hasn't got chaos at all in perspective.

[neutrino86]

I am sorry for my naive venture, but to justify my reasoning, i read in some newscientist article about chaos systems being used to describe fluid motions, which by use of deterministics is profoundly difficult, so it said.

[matt grime]

Chaotic systems are (normally) governed by entirely deterministic systems; there is no probabilisitc input. The 'chaos' appears because of certain properties of these deterministic systems, and not because of any inherent 'randomness'.

 

There is a variant called quantum chaos but I know nothing about that.

[neutrino86]

i see....very intricate nomenclature..

[matt grime]

No, just very misused nomenclature by non-mathematicians.

 

A probabilistic system that displayed some evidence of unpredictability wouldn't exactly be news would it?

[neutrino86]

true...well said.