Probability notation [closed]

Question

I want to ask you about the notation in probability.

I know that

P(A | B) = the conditional probability that event A occurs given that event B has occurred already

But I cannot find what A,B or in my case P(A|B,C). I suggest it means "the conditional probability that event A occurs given that B and C BOTH occurred already"

I don't know what the comma means.

Can you help me ?

Omnipotence

Answer 1

You are basically correct.

P(A| B) is the probability of A given B. P(A| B, C) is the probability of A given (B and C).

You could just as easily write it as P(A| B ∧ C) but it is notational convention to use a comma. Think of everything after the vertical bar as a list of the given things, separated by commas.

(And note that the vertical bar is a very high precedence operator, so to speak.)

Erminna

And is there a notation for "the probability of B or C, givven that A happened": P(A|[B or C]), not P(A,[B and C]) - just curious – Omnipotence

"The probability of B or C, given A" is denoted P(B ∨ C | A). – Deeann

Isn't it lower order of precedence, since you apply the ANDs for the whole list before you apply the given-bar? – Bade

@Bade you are correct, the author of the answer uses the words "high precedence" in the wrong way here. I think what he actually means is "giving the expression structure on a higher level than the commas", which basically means it's evaluated later than the commas, hence it has lower precedence, not higher precedence. – Thusly

@Erminna can you also give an example with many more variables and tell what it means, e.g. P(A, B, C, D, E | F, G, H, I, J, K)? I think it gets clearer then. – Thusly

Answer 2

This is according to Bayes rule

P(C|A,B) = P(A,B|C).P(C) / P(A,B)

Heartsome