Conditional Probability formula

[eprjenkins]

At school we have begun conditional probability. Of course, using the conditional probability formula to answer questions is no problem; but i do not fully understand how the formula works. The formula is;

Pr(A given or │ B)= Pr(A intersection B)/Pr(B)

The the proof for it is self evident when one multiplies both sides by Pr(B).

However, proofs are no use to me if I do not have a feel for why it works. To rid my question of ambiguity: If a mathematician is doodling on some paper and gets: Pr(A intersection B)=(Pr(A intersection B)*Pr(B))/Pr(B) then changes it into Pr(A│B)=Pr(A intersection B)/Pr(B) how will he/she know that Pr(A│B) means probability of A given B? How will they know that this formula pertains to something?


I fear my question may still be steeped in ambiguity...sorry if it is.

 

[HallsofIvy]

I'm not at all clear on what would give YOU a feel for "why it works". "Conditional probability" is really a matter of definition.

Two events are "independent" if and only if P(A and B)= P(A)P(B)- that is, the probability of BOTH happening (your "A intersection B") is just the product of the probability of each happening separately. If two events are NOT independent, then we define the conditional probability P(A|B) (probability that A happens given that B happens) as P(A and B)/P(B).

"If a mathematician is doodling on some paper and gets: Pr(A intersection B)=(Pr(A intersection B)*Pr(B))/Pr(B) then changes it into Pr(A│B)=Pr(A intersection B)/Pr(B) "
Why would he/she do that? They are not at all the same thing. The first is just basic algebra: X= X*Y/Y, while the other is a definition.

"how will he/she know that Pr(A│B) means probability of A given B?"
?? The same way he knows that "3+ 2" means to add! He learned what the symbols mean! Pr(A|B) means the probability of A given B!

"How will they know that this formula pertains to something?" Presumably because he got the formula out of a particular context that referred to something.

 

[ZioX]

Conditional probability is usually a huge conceptual hurdle for someone studying probability. The key to understanding it is to look at examples to motivate how conditional probability should be defined.

Some textbooks will just throw the definition at you and just leave it at that. You'll find yourself wondering why you're even bothering with such things. But you should concern yourself with them because they're really handy tools in modern probability.

David William's chapter on conditional probability is the best treatment on conditional probability I've ever seen. The book itself exceptionally written. See if you can find it in your library and look at the chapter on conditional probability. His construction of them start off with elementary examples and then he'll build his axioms very intuitionally.

https://www.amazon.com/dp/0521406056/?tag=pfamazon01-20

Edit: Hmm, unfortunately I thought this book had a special appendix on conditional probability. It only has a chapter on conditional expectation. Ignore.

 

[eprjenkins]

Ziox and HallsofIvy. Thank you very much for helping me out there, i understand my question was a bit obscure.

HallsofIvy; I think my anecdote describing the mathematician was misunderstood; i will try and change it a little but if you see it as the same story don't worry. I am talking this mathematician is doodling before conditional probability has been invented. He makes up the sign: Pr(A|B) and then equates it to something (Pr(A intersection B)/Pr(B)). How will he know that this equation will mean A given B. You cleared it up a lot more, however, by linking it to the independent thing. Or is it that the a given b law is so fundamental to probability that the mathematician would have to know about it before he could begin doodling?

Edit: Don't worry, I understand now. Because B is given it become the universal set and therefore the total nmber of possible outcomes and A intersection B is (when we are looking for A) the set of successful outcomes!
Much better now.
Ziox: I will see if i can find the book. I realize that conditional probability could be most important and fun; maybe in quantum mechanics? Thankyou.

 

[sjvinay]

solve some problems

The best way to understand conditional probability is solving problems rather than figuring out definitions.

solve 10 or 15 problems, and all your doubts will get cleared.

A good book: Intro to prob by J laurie snell