Algebraic proofs for probability equations?

[CuriousBanker]

Hello

So I'll be going back to school for math next semester, so I might not know the answer to this because I haven't taken combinatorics. I only really know algebra one and two, calc one and some trig.

Anyway, currently studying for my cfa, and it's easy enough to plug the formulas in, but they don't show you how they were derived. So for the two following formulas:

1) P(ab) = p(a) x p(b) (like the chance of getti two heads on two consecutive coin flips
2) p (a|b) = p(ab)/p(b) (conditional probability)

They both make intuitive sense and I can prove them wih a tree diagram but I am just curious if there's an algebraic proof

Thanks I advance, probably a simple answer

 

[mathman]

1) can't be proven as such. Essentially it is an equation describing the relation between independent events. If a and b are not independent, then the equation does not hold.

2) is (almost) the definition of conditional probability.

 

[CuriousBanker]

1) why can't it be proven? And yeah I'm talking about independent events

2) I know it's the definition, but how was it derived?

Can neither be proven then? Sorry if I'm missing something

 

[micromass]

1) why can't it be proven? And yeah I'm talking about independent events

Well, what's your definition of "independent"? As long as you haven't given a rigorous definition, we can't prove anything.

2) I know it's the definition, but how was it derived?

Do you mean "Why did we choose this as the definition?" That is a good question. I think it's because it's "intuitively clear". For example, look at several examples. Those examples are things that we can verify with tree diagrams. So it makes sense to make it a general definition.

 

[CuriousBanker]

Yeah, it is intuitively clear and I can show examples...I just wasn't sure if it was provable using general laws of math (or whatever you call the twelve properties spivak listed )

 

[micromass]

Yeah, it is intuitively clear and I can show examples...I just wasn't sure if it was provable using general laws of math (or whatever you call the twelve properties spivak listed )

Let's say we want to prove (2). In (2), we are given a function $P(\cdot~|~\cdot)$ which can take in two variables. In order to prove something about that function, we must know something about it. So it has to be defined in some way.
So, once you give a rigorous definition of $P(A~|~B)$, we can see if we can prove (2).

 

[CuriousBanker]

Do you have a rigorous definition of two? Is there one? If there is, that's what I'm missing. I'm assuming that mathematicians don't just accept this equation without a rigorous definition. Am I just in over my head? Like should I wait until I start taking probability classes I'm a few years before asking this kind of thing?

 

[micromass]

Do you have a rigorous definition of two? Is there one? If there is, that's what I'm missing. I'm assuming that mathematicians don't just accept this equation without a rigorous definition. Am I just in over my head? Like should I wait until I start taking probability classes I'm a few years before asking this kind of thing?

Yes, there is a rigorous definition of them. What mathematicians do is to take (1) and (2) as definitions. We do that because we can't find anything "simpler" to take as definition.

 

[CuriousBanker]

Yes, there is a rigorous definition of them. What mathematicians do is to take (1) and (2) as definitions. We do that because we can't find anything "simpler" to take as definition.

Would you care to share them with me please?

 

[micromass]

Would you care to share them with me please?

Share what?

 

[CuriousBanker]

The rigorous definitions

 

[micromass]

The rigorous definitions

You have already provided them in your OP.

Definition: $A$ and $B$ are defined to be independent if $P(A\cap B) = P(A)P(B)$.

Definition: if $P(B)\neq 0$, then we define $P(A~\vert~B) = \frac{P(A\cap B)}{P(B)}$.

 

[CuriousBanker]

But how do we know that it will hold for all values, without doing a tree diagram for every possible example? Or do we not know?

 

[micromass]

But how do we know that it will hold for all values, without doing a tree diagram for every possible example? Or do we not know?

We know because we have defined it that way. There is no need to prove a definition.

 

[CuriousBanker]

We know because we have defined it that way. There is no need to prove a definition.

I must not be understanding...so because somebody defined a definition that way, nature conformed around that guys definition? So the chance of flipping a coin two consecutive times and getting two consecutive heads is 25% because somebody defined it as so? What if he made the deviation p(ab)=(p(a) times p(b))^2, would coins start only landing on heads twice in a row 6.25% of the time now since I defined it that way? I know what I am saying sounds ridiculous, but I'm phrasing it this way because maybe now you can see my confusion

 

[micromass]

I must not be understanding...so because somebody defined a definition that way, nature conformed around that guys definition? So the chance of flipping a coin two consecutive times and getting two consecutive heads is 25% because somebody defined it as so? What if he made the deviation p(ab)=(p(a) times p(b))^2, would coins start only landing on heads twice in a row 6.25% of the time now since I defined it that way? I know what I am saying sounds ridiculous, but I'm phrasing it this way because maybe now you can see my confusion

Yes, I do see your confusion, but it's difficult to explain.

First of all, you need to understand that what nature says or thinks doesn't matter for mathematics. So if we defined $P(A\cap B) = (P(A)P(B))^2$, then that would be mathematically valid.
What matters for mathematicians is that we are given a system of basic rules and definitions, and that we can use those to deduce correctly certain laws.
So when you ask "Please prove the following for me", then you must understand that this is entirely dependent of what you accept as rules and definitions.
In particular, definitions cannot be proven.

However, and this is my second point, definitions can be motivated! While defining
$$P(A\cap B) = (P(A)P(B))^2$$
is entirely possible mathematically, this would be a useless definition since it wouldn't conform to what we know about nature. So nobody uses this definition.
However, the definition
$$P(A\cap B) = P(A)P(B)$$
does conform to nature, which is why we use this.

But then you ask of course, "can you prove that it conforms to nature?" No, we can't. A "proof" is something mathematical, and in order to prove something we need to have defined it. Nature does not provide us with a definition of "independence". It only supplies us with some kind of intuition about what it should be.

So while we can never quite prove that definitions conform to nature, we can motivate the definition. We can use our intuition on basic examples and using this, we can see intuitively what should be true. So if we take two dice and throw them separately, then we can see intuitively (using tree diagrams or whatever) that
$$P(A\cap B) = P(A)P(B)$$
This motivates us to say that the above equation should be true for independent events. So we take it as a definition.

As you see, there are two parts in mathematics. You have the formal part where everything is defined for you and you "only" need to deduce things from the rules and definitions. However, many teachers and many students tend to forget about the second part which is the intuitive part. This part asks us to motivate the definitions and rules we work with. Personally, I think this second part is the hardest part, and the most important one.

 

[CuriousBanker]

Ah, thanks for that! Yeah, I kind of already knew that, but I remember asking in another thread about something like that, and somebody told me everything could be proved...and I got confused there.

So basically, somebody played around with a bunch of examples, saw this was the general pattern, and said "ok, let's try to deduce more things using this pattern as the basis until somebody proves it wrong"?

 

[micromass]

Ah, thanks for that! Yeah, I kind of already knew that, but I remember asking in another thread about something like that, and somebody told me everything could be proved...and I got confused there.

Everything can be proven as long as you're given the relevant rules and definitions. Otherwise, a proof is impossible.

So basically, somebody played around with a bunch of examples, saw this was the general pattern, and said "ok, let's try to deduce more things using this pattern as the basis until somebody proves it wrong"?

That's basically it, yes.

 

[CuriousBanker]

Everything can be proven as long as you're given the relevant rules and definitions. Otherwise, a proof is impossible.



That's basically it, yes.

Well yeah, but at least one thing has to be unproven right? There's got to be at least one definition at the stem. Is that the ultimate goal of math to boil everything down to one single definition that everything else can be derived from?

Man I really have a long road ahead of me

 

[micromass]

Well yeah, but at least one thing has to be unproven right? There's got to be at least one definition at the stem. Is that the ultimate goal of math to boil everything down to one single definition that everything else can be derived from?

Man I really have a long road ahead of me

Sure, there will always be things which remain unproven.

But no, I don't think we want to have only one definition where the rest follows from. That would be impossible.

What we want is to take the least amount of axioms and definitions and derive the rest of math from that. The axioms and definitions should however be properly motivated by what happens in nature. So

 

[CuriousBanker]

Makes sense. Thank you very much