Okay, let’s consider the shop floor layout on the screen and each machine on that shop
floor operates with a known probability that you can see. What’s the probability that
throughput can get through the shop floor, okay? That is what’s the probability that
we can start on one end of the shop floor and get through to the other end? So let’s
think about this in a little bit of a different way. Let’s consolidate that picture down
to let’s call it the top and the bottom. Okay, so we have two different ways to get
through the machine floor, okay. Let’s just refer to that string as the top and that string
down there as the bottom. Okay? And let’s identify some events. Let’s call event T
as the top works, that is all of the machines in their work. Remember this would be a series
system along the top, okay. So what’s the probability that we can get through the top?
That’s this event. And let’s identify event B as bottom works. Okay, that is we
can get through put from here through the bottom and out the other side. Okay, therefore
the compliment of T would be the top doesn’t work and the compliment of B, B compliment
would be the bottom doesn’t work. Okay? So let’s identify what’s the probability
that the top works? Okay, for the top to work that means that all of those machines, those
three machines in series have to operate, okay, so what’s that probability? For the
top that would be point nine times point eight, times point seven. What’s the probability
that we can start here, go through the top set of machines and then come out the other
side? That’s that probability, which is zero point five zero four. Okay, therefore
the probability of the compliment of the top works, okay. The top does not work would be
one minus point five zero four which is zero point four nine six. Okay, now let’s find
what’s the probability that the bottom works, okay. Probability that the bottom works is
point nine five times point nine five times point nine five. Each one of those has the
probability of working of point nine five. That, if we work that out is zero point eight
five seven, zero point eight five seven. Okay, so the probability of B compliment would be
one minus point eight five seven which is zero point one four three. Okay so now we
have a new diagram here, a simplified version of what we see on the screen. And now we want
to know what’s the probability that throughput can get through the shop floor. Okay, so what
are the ways that we can get from here to here? Well if either of these work we’re
good, okay. So that’s one way. So the probability that the top works and the bottom works that
event top and bottom will get us through. If the bottom doesn’t work and the top does,
or the top works and the bottom does not, okay? We can still get through and or or,
excuse me, the probability that the top does not work but the bottom does. If the top does
work we still have this option to get through. So top doesn’t, bottom does, okay. So this
probability, okay, if we find this we can find what’s the probability that this thing,
that we can get throughput through this shop? Okay. So let’s count these things individually.
The probability of top and bottom is, these things work independently, let’s assume.
That would be point five zero four times B, probability of B, point eight five seven.
Okay, and that is equal to, well let’s not, let’s not calculate these intermediately.
The probability of top and B compliment, that would be point five oh four and B compliment,
point one four three. And let’s look at the probability of T compliment and B. Okay,
T compliment is point four nine six and B, point eight five seven. Okay, this gives us
this probability over here. The probability that we can get through the shop floor, if
we do all these individual multiplications and sum them up we should get the value zero
point nine two nine, okay? So given these individual component probabilities of operation
what’s the probability that we can get from one side of the shop floor to another that’s
point nine two nine. Another way to think about that is what’s the compliment of this
thing not working at all. If we calculate what’s the probability that both of these
options don’t work, take the compliment of that, that gives the probability that it
does work, okay. So if we find the probability of T compliment and B compliment, take one,
one minus that probability that would give us the probability of this essentially. The
probability that we can get through from one side to the other, okay that’s equal to
one minus the probability of T compliment times the probability of B compliment which
is one minus the probability of T compliment which was point four nine six. Times the probability
of B compliment, point one four three, if we do that we should get one minus point zero
seven one which is equal to zero point nine two nine. Okay, so same way to get these two
answers and as it turns out, so what this is, these are two sub systems that are in
parallels and the sub systems are in series. Okay, so we have a combination of series in
parallel in this shop floor system.