Re: conditional Probabilities help
From: shedar (nobody_at_nonesuch.com)
Date: 09/26/04
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Date: Sun, 26 Sep 2004 09:31:34 GMT
"nsgi_2004" <nospam@nospam.com> wrote in message
news:ons5d.35176$aW5.43@fed1read07...
>
> "shedar" <nobody@nonesuch.com> wrote in message
> news:ERq5d.107361$%S.87064@pd7tw2no...
> > "nsgi_2004" <nospam@nospam.com> wrote in message
> > news:hVp5d.35144$aW5.16749@fed1read07...
> > > Hello,
> > >
> > > The conditional probability defined:
> > >
> > > P( E | F ) = P( EF ) / P(F), where EF means E intersects F.
> > >
> > > It is not clear to me how this equation is derived.
> > >
> > > My book says: "...because we know that F has occurred, it follows that
F
> > > becomes our new sample space and hence the probability that the event
EF
> > > occurs will equal the probability of EF relative to the probability of
> F."
> > >
> > > I understand how F is the new sample space because we know F occured.
> > > However, I don't get the second part of the above sentence--the
"hence"
> > > part.
> > >
> > >
> >
> > Let S denote the sample space, and let E and F be subsets of S.
> >
> > Notation: "EF" denotes the intersection of E and F.
> >
> > The equation P(E | F) = P(EF)/P(F) is not derived; rather, it is the
> > official definition of P( E | F ), which is called the (conditional)
> > probability of E given F. It is still a fair question, though, to ask
why
> it
> > is defined like that. Here is one intuitive way to view it:
> >
> > P(E | F) is used to denote the probability of the event E GIVEN THAT the
> > event F has occurred. Under the explicit assumption that F has already
> > occurred, the sample points (aka, outcomes) in the "part" of E that is
> > "outside" of F will not have any contribution to the (further)
probability
> > of E; as a result, we need only compute how many sample points there are
> in
> > the part of E that is also part of F, hence the focus on the
intersection
> of
> > E and F. In this manner, F effectively becomes the "restricted sample
> > space" (think of it as "shrinking the sample space from S down to F").
So,
> > speaking tersely and informally,
> >
> > P(E) (assuming F has occurred) = P(E inside F) / P(restricted
sample
> > space).
> >
> > In formal notations, P(E | F) = P(EF)/P(F).
> >
> > Loosely speaking, it is the ratio of how many sample points there are in
> EF
> > over how many sample points there are in the restricted sample space F
(if
> > one measures probabilities using the "counting measure" as it is usually
> > done when the sample space is finite).
>
> Ok, I think I'm starting to get it. Let me see if I can articulate it.
>
> F is the new sample space.
>
> EF can range from the empty set to complete intersection with F (EF = F).
Yes
> The more EF and F overlap the more likely E is to occur and conversely.
No. I think you meant to say "the more E and F overlap ..." The converse
may not be true, though.
>
> So the ratio P(EF) / P(F) basically gives us a probability number in [0,
1]
> to model that. E.g., EF = F means P(EF) / P(F) = 1,
(EF = F) IMPLIES P(E | F) = P(EF)/P(F) = 1
but the converse does not necessary hold (don't worry about why the converse
may fail at this point; one needs to know some "measure
theory"--specifically, exotic sets called "sets with measure zero"--before
this can be explained).
> which means E certainly
> occurs, which is obviously true if the event = sample space. If EF
> intersects half of F then we would get P(EF) / P(F) = 1/2.
>
The phrase "Half of F" is a bit vague. One needs to specify how sample
points are "counted" or "measured". For a finite sample space, counting the
number of sample points would not be a problem. [But food for thought: how
would one "count" sample points if the sample space or the event is
uncountable? Don't worry about this too much at this point if you are only
taking an introductory (2nd year college) class in probability or
statistics.]
> Although I see that this works, I think what was confusing is why not
define
> it as:
>
> P(E | F) = (num sample point in EF) / (num sample points in F)
>
> I guess because the definition using the probability ratios turns out to
be
> easier to solve problems with. Is that why or is there a deeper reason?
>
Defining it as the "n(EF) divided by the n(F)" would presuppose that one is
using the "counting measure" to compute probabilities: P(A) = n(A) / n(S),
where n(A) is the number of sample points favorable to A, and n(S) is the
total number of sample points in S with S nonempty. This, in turn, would
presuppose that one is able to "count" elements in a set, which, in turn,
would presuppose that the sets are FINITE. In the case of uncountable sets,
the word "count" would be misleading, and one needs a more sophisticated way
of "measuring" them appropriately in the context of computing probabilities.
E.g., randomly pick a number from the open interval S = (-1,1), and let A be
the event {t in S | t < 0}. What is the probability of A? One would desire
this probability to be 1/2. But observe that both A and S are uncountable,
so ordinary counting does not apply.
What level is this probability/statistics class that you are taking (2nd
year, 3rd year, 4th year university)? Is it calculus based?
Shedar
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