Probability notation: question about joint and conditional probability

[Master1022]

Homework Statement

What does [itex] p(a|b, c) [\itex] mean?

Relevant Equations

Joint probability
Conditional probability

Hi,

Just a quick question about conditional and marginal probabilities notation.

Question: What does $p(a|b, c)$ mean?
Does it mean:
1) The probability of A, given (B and C) - i.e. $p[A | (B \cap C)]$ OR

2) The probability of (A given B) and C - i.e. $p[(A | B) \cap C]$

I was also wondering if there was an intuitive way to understand how to break it down, but I should probably try to understand this first.

Thanks

 

[Office_Shredder]

It means the first thing. I don't even know how to interpret the second thing you wrote down.

 

[FactChecker]

Given a joint discrete PDF, p(x,y), I would always interpret p(x,y) as $P((X=x) \cap (Y=y))$. I would interpret p(a|b, c) as $P( (A=a|B=b) \cap (C=c) )$.

CORRECTION: Sorry, I forgot to do the conditional part correctly. I should have said $P( (A=a)\cap (B=b) \cap (C=c) )/P(B=b)$

 

[Master1022]

Thank you very much for both of your replies @Office_Shredder and @FactChecker . However, I am still slightly confused as you both said different things...

Thinking about it again, I think the first option makes sense, but just wanted to confirm.

It means the first thing. I don't even know how to interpret the second thing you wrote down.

Given a joint discrete PDF, p(x,y), I would always interpret p(x,y) as $P((X=x) \cap (Y=y))$. I would interpret p(a|b, c) as $P( (A=a|B=b) \cap (C=c) )$.

Thanks

 

[PeroK]

Homework Statement:: What does $p(a|b, c)$ mean?

I would say the notation is sufficiently unusual to need a specific note in the text on how to interpret it.

 

[Office_Shredder]

I don't understand what Fact Checker's answer means. What does it mean for A=a given B=b and for C=c unless we are assuming B=b for that second part also?

Like suppose we have two coins we flip. What is P(both coins are heads| first coin is heads, second coin is heads). How do you compute it?