>> While judging from the discussions on the on the cost problem in problem set one
the questions that caused people the most difficulty were numbers 6 and 7.
Those are those questions about Alice, and let's just see, see what's going on here.
Notice that in each of these, we've got the same phrase, works in a bank, works in
a bank, works in a bank, works in a bank, works in a bank.
But for 4 of these, there's an additional requirements.
Not only does she work in a bank, but something else, works in a bank, something
else. Works in a bank, something else.
Works in a bank, something else. Each of these additional requirements.
Each time you got 1, it makes it less likely to happen.
Because you've got something else to satisfy.
They all have to satisfy the fact that Alice works in a bank.
But for the first 4, Alice also has to satisfy something else.
So each of these. Makes it less likely, that, that makes it
less likely to be true. So this one is the one most likely to be
true. Now it's possible that in a particular
instance another one has the same likelihood, but not saying that there's a
unique most likely one. But we are seeing is the most likely one.
Incidentally it doesn't matter whether you express in terms of the word likely or the
word probable or probability. You can do it numerically.
You can do it however you want. The issue is not what the word is.
The issue is the information you have. And in each of these cases, you have an
extra restriction and whenever you got an extra restriction with a conjunction, and
it's critical that was conjunction here, when you go next to restriction with a
conjunction, it makes it less likely, because instead of just having to satisfy
one thing, you have to satisfy two things. When you satisfy two things, it makes it
less likely to happen. So we got to see if whether you'll use the
word likely, probable, whatever you want to do The most likely one is this one,
because this is the easiest one to satisfy.
And this right here I think is a great example of mathematical thinking.
I mean, this is a superb example of the kind of thing that I'm talking about in
this course, because we're arguing purely in terms of mathematics.
In fact, It's the mathematics of the word and the reason is, is in the problem set,
is because this is all about where and works.
So if you really understand how and works, this one just pops right out.
That's why it's there. It's, it's to, to make you really reflect
on, on conjunction, on the power of conjunction.
Okay? So it's there you go.
It, it's also do with the amount of information you get.
Incidentally, all of probability theory, does not say all, I mean, probability
theories are powerful theory, but what probability theory does, is adds numbers,
to the amount of information we have. Okay?
And in this case, the extra information is more restrictive.
And, and, and when you assign probabilities to that, that gets
reflected. Okay.
Well that was number 6. Let's take a look at number 7.
Number 7 is slightly different, because we don't have a common expression in, in all
of the, all of the five. In the case works in the bank.
In this one we have works in a bank and something else.
This one I've got is a rock star, this one I've got works in a bank and something
else, this one we've got works in a bank and here we've got works in a bank.
Okay, the other thing here is being quiet, being honest and so forth.
But if you look through these, this one has Works in a bank, and something else.
So this is less likely than working in a bank.
This is being a rock star. This is honest,[INAUDIBLE].
This one is also less likely to work in a bank.
So if we're looking for the most likely thing, these make them less likely.
So let's just sort of ignore that part. That makes it less likely.
It's already more likely that she works in a bank.
This one is nothing to be more. This one she works in a bank.
So we've got either working in a bank. You know, this is more likely than it was
originally so I'm making things more likely.
Here we've got Alice being a rock star. In this one, either she's a rock star or
she works in a bank. So this is the most general one.
This is more likely. Because there are two chances here, either
working in a bank or being a rock star. Here there's just one of them.
Here there's just one of them. Here's one of them, and here there's one
of them. Okay?
So the most likely one is this one, because of disjunction.
There were two ways that she could satisfy this, by being a rock star and working in
a bank. In each of these There's one of them, and
I've ignored the thing that makes it even less likely.
So having modified that I've made B more likely.
Having modified D I've made it more likely.
So no I've just got works in a bank, works in a bank, rock star, works in a bank, but
there's only one of them in each of that. In this one there are two of them.
So again, purely By using mathematical thinking I've arrived at the one's most
likely. And as in number six, it doesn't matter if
you take expressed in terms of probabilities or relative probabilities.
However you want internetic probabilities are in many different types of
probability. There's frequencies probability, there's
Beijing probability, there's subjective probability, there's epistemic
probability. Assigning numbers to the information you
have and calling them probabilities is a pretty tricky thing to, there' are many
different ways of doing it, It. And it's, it's just a very complicated
issue, so those of you who tried to use probability theory, that's fine, you can
do it. But it actually gets more complicated not
least because you end up having to talk about whether things are independent
events and so forth. And the point is you don't need any of
that you know, remember at the beginning I said, this course is not about Rushing in
and applying techniques or procedures. It's about thinking basically about the
issues. And you don't need the machinery here.
If you think about the issues, it comes out.
Remember my, my exhortations. Look at a problem, ask yourself what it
means, what it is saying, and try to reason in terms of the problem itself.
Don't look for techniques to apply. You often need to do that, but in this
case that's not what the cause is about. And many of the questions I'm giving in
this course should be solved not by applying a technique, but by, by just
thinking about the problem. And, and let me finish this, this, this,
tutorial on the problem section. Now, these are the only two problems we
want to look at. Let me finish that.
This, this tutorial section, with a different problem, of the same kind where
you can solve it by just thinking about the problem.
Not applying a technique. And here is that problem.
Okay? You've arrived late at an airport.
You're rushing to catch your plane. Unfortunately, your gate is at the far end
of the terminal. It always seems to be at the far end of
the terminal, doesn't it? The fastest you can walk is a constant
speed of four miles per hour. For part of the way, there is a moving
walkway moving at two miles per hour. You decide to take the walkway and
continue to walk. So you're going to walk all the way, but
part of, when there's a walkway you're going to walk on the walkway so you can go
faster, right? Just as you're about to step onto the
walkway, you notice your shoelace is loose.
And the last thing is you want to do is get your shoelace caught in the mechanism
of the walkway. So you've got a choice.
You either stop and fasten your shoelace just before you step on the walkway, or
you step on the walkway and then stop to fasten your shoelace whilst the walkway is
moving you along. And there's no other options.
Those are the two options you have. Which option will get you to the gate
fastest? Now there are 2 ways you could try to go
about this. You could try to argue in terms of speed,
adding the speeds together and so forth, and if you do that you're going to run
into a problem. There's going to be some information that
you need that you don't have. For example, I haven't told you how long
the walkway is and how long have to walk in general.
So there's, there's, there's a problem if you try to use stunted methods.
However, you do have enough information to solve the problem.
But if you try to apply relative speed to the kind of methods you were taught to use
in, in high school, for solving problems about speed, relative speed, and so forth
you're going to run into problems, and you don't need that.
If you think about the problem, you should be able to solve it.
Well, I'm just going to leave you with that one.
It's an interesting problem, I think it's a rather neat problem.
There are many problems like this that, that you can really can just solve by
mathematical thinking. And the point is, this again is all about
this important notion Mathematical thinking.
It's powerful, if you can master mathematical thinking many problems can be
resolved without going into the complexity of applying mathematical techniques.
Mathematical techniques are what you need when mathematical thinking alone, doesn't
work. But often you can get by with just
mathematical thinking so, good luck on that one and enjoy it it's a lot of fun.
Dr. Keith DevlinCo-founder and Executive Director
