What is the Meaning of P(E given F) in Basic Probability?

[Redd]

I have never been very good at probability, and I am confused with this rather simple statement:

"BASIC FACT:
Let E be any event, and F and G be events such that one and only one of the events F and G will occur. Then

P(E) = P(F)*P(E given F) + P(G)*P(E given G)"

Where P(E) = the probability of E occurring. And the same for the others.

To be honest I don't even understand what it is asking me to do procedurally. What does it mean "E given F"? Is that the probability of E occurring if F occurs? Why is that pertinent? More than that I don't understand the reasoning nor do I have any intuitive inkling as to why this expression would yield the correct answer.
Can someone give an example perhaps?
(The book I was given just assumes the reader automatically understands this property).

Any help would be greatly appreciated.

 

[mathman]

The best way to understand it is by using a Venn diagram. The descriptions of F and G are such that they don't overlap and together fill the entire event space. Place E on the diagram and you see that part of it may overlap F while the rest would overlap G.

To add things up properly, P(E given F) means the area of the part of E overlapping F divided by the area of F. Similarly for P(E given G). To get the E area, multiply each piece by the area of F or G as needed.

 

[Redd]

The best way to understand it is by using a Venn diagram. The descriptions of F and G are such that they don't overlap and together fill the entire event space. Place E on the diagram and you see that part of it may overlap F while the rest would overlap G.

To add things up properly, P(E given F) means the area of the part of E overlapping F divided by the area of F. Similarly for P(E given G). To get the E area, multiply each piece by the area of F or G as needed.

Okay. I still have a couple questions.
If P(F) and P(G) fill the "entire event space" does that mean P(F) + P(G) always = 1?
And is that just because one and only one of the events must occur?
That seems to make sense.
So E is dependent on F and G, and this description is finding the probability of E as it depends on the outcomes of F or G?
I hope I'm not misunderstanding because it seems to fit now.

On a side note, do you know if there is a name for this sort of thing so that I can look into it more, or should I just look into general probability basics?
Thanks :)

 

[SW VandeCarr]

Okay. I still have a couple questions.
If P(F) and P(G) fill the "entire event space" does that mean P(F) + P(G) always = 1?

Yes

So E is dependent on F and G, and this description is finding the probability of E as it depends on the outcomes of F or G?

P(F v G)=1; P(F^G)= 0; P(F) = 1 - P(G); P(G)= 1 - P(F).

On a side note, do you know if there is a name for this sort of thing so that I can look into it more, or should I just look into general probability basics?

These types of problems are about the most basic probability examples, such as coin tosses with fair or biased coins. I guess you could call them strict dichotomies.