Previously, we learned the concept of independent
experiments.
In this exercise, we'll see how the seemingly simple idea
of independence can help us understand the behavior of
quite complex systems.
In particular, we'll combined the concept of independence
with the idea of divide and conquer, where we break a
larger system into smaller components, and then using
independent properties to glue them back together.
Now, let's take a look at the problem.
We are given a network of connected components, and each
component can be good with
probability P or bad otherwise.
All components are independent from each other.
We say the system is operational if there exists a
path connecting point A here to point B that go through
only the good components.
And we'd like to understand, what is the probability that
system is operational?
Which we'll denote by P of A to B.
Although the problem might seem a little complicated at
the beginning, it turns out only two
structures really matter.
So let's look at each of them.
In the first structure, which we call the serial structure,
we have a collection of k components, each one having
probability P being good, connected one next to each
other in a serial line.
Now, in this structure, in order for there to be a good
path from A to B, every single one of the
components must be working.
So the probability of having a good path from A to B is
simply P times P, so on and so, repeated k times, which is
P raised to the k power.
Know that the reason we can write the probability this
way, in terms of this product, is because of the
independence property.
Now, the second useful
structure is parallel structure.
Here again, we have k components one, two, through
k, but this time they're connected in parallel to each
other, namely they start from one point here and ends at
another point here, and this holds for
every single component.
Now, for the parallel structure to work, namely for
there to exist a good path from A to B, it's easy to see
that as long as one of these components works the whole
thing will work.
So the probability of A to B is the probability that at
least one of these components works.
Or in the other word, the probability of the complement
of the event where all components fail.
Now, if each component has probability P to be good, then
the probability that all key components fail is 1 minus P
raised to the kth power.
Again, having this expression means that we have used the
property of independence, and that is probability of having
a good parallel structure.
Now, there's one more observation that will be
useful for us.
Just like how we define two components to be independent,
we can also find two collections of components to
be independent from each other.
For example, in this diagram, if we call the components
between points C and E as collection two, and the
components between E and B as collection three.
Now, if we assume that each component in both
collections--
they're completely independent from each other, then it's not
hard to see that collection two and three behave
independently.
And this will be very helpful in getting us the breakdown
from complex networks to simpler elements.
Now, let's go back to the original problem of
calculating the probability of having a good path from point
big A to point big B in this diagram.
Based on that argument of independent collections, we
can first divide the whole network into three
collections, as you see here, from A to C, C to E and E to
B. Now, because they're independent and in a serial
structure, as seen by the definition of a serial
structure here, we see that the probability of A to B can
be written as a probability of A to C multiplied by C to E,
and finally, E to B.
Now, the probability of A to C is simply P because the
collection contains only one element.
And similarly, the probability of E to B is not that hard
knowing the parallel structure here.
We see that collection three has two components in
parallel, so this probability will be given by 1 minus 1
minus P squared.
And it remains to calculate just the probability of having
a good path from point C to point E. To get a value for P
C to E, we notice again, that this area can be treated as
two components, C1 to E and C2 to E connected in parallel.
And using the parallel law we get this probability is 1
minus 1 minus P C1 to E multiplied by the 1 minus P C2
to E. Know that I'm using two different characters, C1 and
C2, to denote the same node, which is C. This is simply for
making it easier to analyze two branches where they
actually do note the same node.
Now P C1 to E is another serial connection of these
three elements here with another component.
So the first three elements are connected in parallel, and
we know the probability of that being successful is 1
minus P3, and the last one is P.
And finally, P C2 to E. It's just a single element
component with probability of successful being P. At this
point, there is no longer any unknown variables, and we have
indeed obtained exact values for all the quantities that
we're interested in.
So starting from this equation, we can plug in the
values for P C2 to E, P C1 to E back here, and then further
plug in P C to E back here.
That will give us the final solution, which is given by
the following somewhat complicated formula.
So in summary, in this problem, we learned how to use
the independence property among different components to
break down the entire fairly complex network into simple
modular components, and use the law of serial and parallel
connections to put the probabilities back together in
common with the overall success probability of finding
a path from A to B.