Independence is a very useful property.
Whenever it is true, we can break up complicated
situations into simpler ones.
In particular, we can do separate calculations for each
piece of a given model and then combine the results.
We're going to look at an application of this idea into
the analysis of reliability of a system that consists of
independent units.
So we have a system that consists of a number, let's
say, n, of units.
And each one of the units can be "up" or "down".
And it's going to be "up" with a certain probability pi.
Furthermore, we will assume that unit failures are
independent.
Intuitively, what we mean is that failure of some of the
units does not the affect the probability that some of the
other units will fail.
If we want to be more formal, we might proceed as follows.
We could define an event Ui to be the event that
the ith unit is "up".
And then make the assumption that the events U1, U2, and so
on up to Un, if we have n units, are independent.
Alternatively, we could define events Fi, where event Fi is
the event that the ith unit is down, or that it has failed.
And we could assume that the events Fi are independent, but
we do not really need a separate assumption.
As a consequence of the assumption that the Ui's are
independent, one can argue that the Fi's are also
independent.
How do we know that this is the case?
If we were dealing with just two units, then this is a fact
that we have already proved a little earlier.
We did prove that if two events are independent, then
their complements are also independent.
Now that we're dealing with multiple events here, a
general number n, how do we argue?
One approach would be to be formal and start from the
definition of independence of the U events.
And that definition gives us a number of formulas.
Then manipulate those formulas to prove the conditions that
are required in order to check that the events Fi are
independent.
This is certainly possible, although it is a bit tedious.
However, the approach we will be taking in situations like
this one is that we will use the intuitive understanding
that we have of what independence means.
So independence in this context means that whether
some units are "up" or down, does not change the
probabilities that some of the other units
will be "up" or down.
And by taking that interpretation, independence
of the events that units are "up" is essentially the same
as independence of the units [having]
failed.
So we take this implication for granted and now we move to
do some calculations for specific systems.
Consider a particular system that consists of three
components.
And we will say that the system is "up", if there
exists a path from the left to the right that consists of
units that are "up".
So in this case, for the system to be "up", we need all
three components to be "up" and we proceed as follows.
The probability that the system is "up"--
this is the event that the first unit is "up", and the
second unit is "up", and the third unit is "up".
And now we use independence to argue that this is equal to
the probability that the first unit is "up" times the
probability that the second unit is "up" times the
probability that the third unit is "up".
And in the notation that we have introduced this is just
p1 times p2 times p3.
Now, let us consider a different system.
In this system, we will say that the system is "up",
again, if there exists a path from the left to the right
that consists of units that are "up".
In this particular case the system will be "up", as long
as at least one of those three components are "up".
We would like again to calculate the probability that
the system is "up".
And the system will be "up", as long as either unit 1 is
"up", or unit 2 is "up", or unit 3 is "up".
How do we continue from here?
We cannot use independence readily, because independence
refers to probabilities of intersections of events,
whereas here we have a union.
How do we turn a union into an intersection?
This is what De Morgan's Laws allow us to do, and involves
taking complements.
Instead of using formally De Morgan's Laws, let's just
argue directly.
Let us look at this event.
That unit 1 fails, and unit 2 fails, and unit 3 fails.
What is the relation between this event and the event that
we have here.
They're complements.
Why is that?
Either all units fail, which is this event, or there exists
at least one unit, which is "up".
So since this event is the complement of that event, this
means that their probabilities must add to 1, and therefore
we have this relation.
And now we're in better shape, because we can use the
independence of the events F to write this as 1 minus the
product of the probabilities that each one
of the units fails.
And with the notation that we have introduced using the
pi's, this is as follows.
The probability that unit 1 fails is 1 minus the
probability that it is "up".
Similarly, for the second unit, 1 minus the probability
that it is "up".
And the same for the third unit.
So we have derived a formula that tells us the reliability,
the probability that a system of this kind is "up" in terms
of the probabilities of its individual components.
You will have an opportunity to deal with more examples of
this kind, a little more complicated, in the problem
that follows.
And even more complicated, in one of the problem-solving
videos that we will have available for you.