# Probability notation: question about joint and conditional probability

• Master1022
The probability of both coins being heads is ##P( (both coins are heads) \cap (first coin is heads) )##.f

#### 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

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) )##.

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)##

Last edited:
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

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.

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