Conditional Probability

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jpanknin

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I'm probably missing the definition of this, but I can't seem to find it anywhere in the book. Can someone please explain the notation [imath]P(C|A, B)[/imath]? I'm familiar with [imath]P(B|A)[/imath] notation (Probability of A given that B has occurred), but I don't understand the first notation. Is it Probability of C given (A and B), Probability of C given A and the B means something else? Thanks in advance.

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jpanknin said:
I'm probably missing the definition of this, but I can't seem to find it anywhere in the book. Can someone please explain the notation [imath]P(C|A, B)[/imath]? I'm familiar with [imath]P(B|A)[/imath] notation (Probability of A given that B has occurred), but I don't understand the first notation. Is it Probability of C given (A and B), Probability of C given A and the B means something else? Thanks in advance.

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What book is this in? Can you show us the context? I suspect you'd find some clues there.

I think it means just what you'd think it means: the probability of event C, given that events A and B happen.
 
Dr.Peterson said:
What book is this in? Can you show us the context? I suspect you'd find some clues there.

I think it means just what you'd think it means: the probability of event C, given that events A and B happen.
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It's "Introduction to Probability, Statistics, and Random Processes" by Pishro-Nik. There's an online version and this is the section on conditional probability (https://www.probabilitycourse.com/chapter1/1_4_0_conditional_probability.php). If you scroll down the page to equation 1.7 you'll see this notation.

Our class textbook is Devore, "Probability and Statistics," 9E, but I find Pishro-Nik much easier to follow with less terse introductions. I don't see any of the comma notation in Devore's book. The image below from Devore's section on conditional probability shows what I think is the same thing in the first image in the OP. Where instead of the comma notation used in the original post [imath]P(C|A, B)[/imath], Devore uses an intersection. Unless these are different. Not sure.

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Yes, I think that's the same idea; I would use the intersection, where the one author uses the comma. I've seen the latter in some places.

Clearly the former book is saying the same thing as your image here:

1722860919516.png​

It would be nice if the notation were explicitly defined, but they are implicitly doing so here.
 
Looks to me that [imath]P(C|A,B)[/imath] is the same as [imath]P(C| A\cap B)[/imath]. At least the equality would hod with such definition.
 
Dr.Peterson said:
Yes, I think that's the same idea; I would use the intersection, where the one author uses the comma. I've seen the latter in some places.

Clearly the former book is saying the same thing as your image here:


It would be nice if the notation were explicitly defined, but they are implicitly doing so here.
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Ok, thank you. And in terms of grouping (order of operations) would it be [imath](C | A ) \cap B[/imath] or [imath]C | (A \cap B)[/imath]?
 
jpanknin said:
Ok, thank you. And in terms of grouping (order of operations) would it be [imath](C | A ) \cap B[/imath] or [imath]C | (A \cap B)[/imath]?
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Neither of those has any meaning by itself! A, B, and C are events; "given that" is not an operation on events.

I'll take your question as "would [imath]P(C | A \cap B)[/imath] be [imath]P((C | A ) \cap B)[/imath] or [imath]P(C | (A \cap B))[/imath]?"

The latter represents the intended meaning. The vertical bar is, again, not an operation that produces an event, but a divider between two parts of the description of what probability we want: P( event | event) is the probability of this event given that event, so that everything after the bar is the condition.
 
Dr.Peterson said:
Neither of those has any meaning by itself! A, B, and C are events; "given that" is not an operation on events.

I'll take your question as "would [imath]P(C | A \cap B)[/imath] be [imath]P((C | A ) \cap B)[/imath] or [imath]P(C | (A \cap B))[/imath]?"

The latter represents the intended meaning. The vertical bar is, again, not an operation that produces an event, but a divider between two parts of the description of what probability we want: P( event | event) is the probability of this event given that event, so that everything after the bar is the condition.
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Perfect. Thank you very much.
 
Me 2 sikkas ...

[imath]P((A \cap B)|C) \text{ or } P(A|(B \cap C))[/imath] can be "simplified" to (if [imath]A \cap B = D \text{ and } B \cap C = E[/imath]) [imath]P(D|C) \text{ and } P(A|E)[/imath]
 
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