Hello,
I have a question regarding basic probability and statistics. If I
understand correctly, the definition of independence holds for two events
that are subsets of the same sample space. In other cases, we may need to
construct a new sample space, such as with the flipping of a coin twice.
Here, we construct the new sample space S=S1xS2={HH,HT,TH,TT}, where
Si={H,T} for i=1,2. This way, two events that are independent, such as
A="head on first toss" and B="tails on second toss," are subsets of the same
sample space.
Now, the problem that I have is that, while it is not difficult to
construct sample spaces intuitively for textbook problems, it is difficult
to do so using these basic definitions of probability. For ex., consider a
problem where a manufacturer has five seemingly identical computers, though
two are really defective and three are good. An order calls for two of the
computers, and we want the probability of the event A="order is filled with
two good computers."
Intuitively, it is obvious that if D1 and D2 are the bad computers, and
G1-G3 are the good computers, then
S={D1D2,D1G1,D1G2,D1G3,D2G1,D2G2,D2G3,G1G2,G1G3,G2G3}. Thus, P(A)= 0.30.
However, I cannot think of any way of constructing the sample space using
definitions like the cartesian product. Perhaps this is because the second
computer chosen depends on which computer is chosen first. Yet, another
similar problem in my textbook states that the probabilities of a computer
being good and defective (from a particular manufacturer) are 0.90 and 0.10,
respectively. Then, if we want to test five computers, we may construct the
sample space S=S1xS2xS3xS4xS5, where Si={G,D} for i=1,...,5. Hence, if
A="all five computers tested are good," P(A)=(0.90)^5. Why is that we can
use the Cartesian product in this case but not in the other case? Is it that
in the first case we are not performing an experiment, but just sampling?
Perhaps I am thinking about this too much, but it would be nice to be able
to construct these sample spaces for problems using some sort of formulaic
method, as opposed to intuition (perhaps this isn't the right way to view
this subject?). Any help would be greatly appreciated.
_______________________________________________________
Send a cool gift with your E-Card
http://www.bluemountain.com/giftcenter/
=================================================================
Instructions for joining and leaving this list and remarks about
the problem of INAPPROPRIATE MESSAGES are available at
http://jse.stat.ncsu.edu/
=================================================================
