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>> While judging from the discussions on the on the cost problem in problem set one

Â the questions that caused people the most difficulty were numbers 6 and 7.

Â Those are those questions about Alice, and let's just see, see what's going on here.

Â Notice that in each of these, we've got the same phrase, works in a bank, works in

Â a bank, works in a bank, works in a bank, works in a bank.

Â But for 4 of these, there's an additional requirements.

Â Not only does she work in a bank, but something else, works in a bank, something

Â else. Works in a bank, something else.

Â Works in a bank, something else. Each of these additional requirements.

Â Each time you got 1, it makes it less likely to happen.

Â Because you've got something else to satisfy.

Â They all have to satisfy the fact that Alice works in a bank.

Â But for the first 4, Alice also has to satisfy something else.

Â So each of these. Makes it less likely, that, that makes it

Â less likely to be true. So this one is the one most likely to be

Â true. Now it's possible that in a particular

Â instance another one has the same likelihood, but not saying that there's a

Â unique most likely one. But we are seeing is the most likely one.

Â Incidentally it doesn't matter whether you express in terms of the word likely or the

Â word probable or probability. You can do it numerically.

Â You can do it however you want. The issue is not what the word is.

Â The issue is the information you have. And in each of these cases, you have an

Â extra restriction and whenever you got an extra restriction with a conjunction, and

Â it's critical that was conjunction here, when you go next to restriction with a

Â conjunction, it makes it less likely, because instead of just having to satisfy

Â one thing, you have to satisfy two things. When you satisfy two things, it makes it

Â less likely to happen. So we got to see if whether you'll use the

Â word likely, probable, whatever you want to do The most likely one is this one,

Â because this is the easiest one to satisfy.

Â And this right here I think is a great example of mathematical thinking.

Â I mean, this is a superb example of the kind of thing that I'm talking about in

Â this course, because we're arguing purely in terms of mathematics.

Â In fact, It's the mathematics of the word and the reason is, is in the problem set,

Â is because this is all about where and works.

Â So if you really understand how and works, this one just pops right out.

Â That's why it's there. It's, it's to, to make you really reflect

Â on, on conjunction, on the power of conjunction.

Â Okay? So it's there you go.

Â It, it's also do with the amount of information you get.

Â Incidentally, all of probability theory, does not say all, I mean, probability

Â theories are powerful theory, but what probability theory does, is adds numbers,

Â to the amount of information we have. Okay?

Â And in this case, the extra information is more restrictive.

Â And, and, and when you assign probabilities to that, that gets

Â reflected. Okay.

Â Well that was number 6. Let's take a look at number 7.

Â Number 7 is slightly different, because we don't have a common expression in, in all

Â of the, all of the five. In the case works in the bank.

Â In this one we have works in a bank and something else.

Â This one I've got is a rock star, this one I've got works in a bank and something

Â else, this one we've got works in a bank and here we've got works in a bank.

Â Okay, the other thing here is being quiet, being honest and so forth.

Â But if you look through these, this one has Works in a bank, and something else.

Â So this is less likely than working in a bank.

Â This is being a rock star. This is honest,[INAUDIBLE].

Â This one is also less likely to work in a bank.

Â So if we're looking for the most likely thing, these make them less likely.

Â So let's just sort of ignore that part. That makes it less likely.

Â It's already more likely that she works in a bank.

Â This one is nothing to be more. This one she works in a bank.

Â So we've got either working in a bank. You know, this is more likely than it was

Â originally so I'm making things more likely.

Â Here we've got Alice being a rock star. In this one, either she's a rock star or

Â she works in a bank. So this is the most general one.

Â This is more likely. Because there are two chances here, either

Â working in a bank or being a rock star. Here there's just one of them.

Â Here there's just one of them. Here's one of them, and here there's one

Â of them. Okay?

Â So the most likely one is this one, because of disjunction.

Â There were two ways that she could satisfy this, by being a rock star and working in

Â a bank. In each of these There's one of them, and

Â I've ignored the thing that makes it even less likely.

Â So having modified that I've made B more likely.

Â Having modified D I've made it more likely.

Â So no I've just got works in a bank, works in a bank, rock star, works in a bank, but

Â there's only one of them in each of that. In this one there are two of them.

Â So again, purely By using mathematical thinking I've arrived at the one's most

Â likely. And as in number six, it doesn't matter if

Â you take expressed in terms of probabilities or relative probabilities.

Â However you want internetic probabilities are in many different types of

Â probability. There's frequencies probability, there's

Â Beijing probability, there's subjective probability, there's epistemic

Â probability. Assigning numbers to the information you

Â have and calling them probabilities is a pretty tricky thing to, there' are many

Â different ways of doing it, It. And it's, it's just a very complicated

Â issue, so those of you who tried to use probability theory, that's fine, you can

Â do it. But it actually gets more complicated not

Â least because you end up having to talk about whether things are independent

Â events and so forth. And the point is you don't need any of

Â that you know, remember at the beginning I said, this course is not about Rushing in

Â and applying techniques or procedures. It's about thinking basically about the

Â issues. And you don't need the machinery here.

Â If you think about the issues, it comes out.

Â Remember my, my exhortations. Look at a problem, ask yourself what it

Â means, what it is saying, and try to reason in terms of the problem itself.

Â Don't look for techniques to apply. You often need to do that, but in this

Â case that's not what the cause is about. And many of the questions I'm giving in

Â this course should be solved not by applying a technique, but by, by just

Â thinking about the problem. And, and let me finish this, this, this,

Â tutorial on the problem section. Now, these are the only two problems we

Â want to look at. Let me finish that.

Â This, this tutorial section, with a different problem, of the same kind where

Â you can solve it by just thinking about the problem.

Â Not applying a technique. And here is that problem.

Â Okay? You've arrived late at an airport.

Â You're rushing to catch your plane. Unfortunately, your gate is at the far end

Â of the terminal. It always seems to be at the far end of

Â the terminal, doesn't it? The fastest you can walk is a constant

Â speed of four miles per hour. For part of the way, there is a moving

Â walkway moving at two miles per hour. You decide to take the walkway and

Â continue to walk. So you're going to walk all the way, but

Â part of, when there's a walkway you're going to walk on the walkway so you can go

Â faster, right? Just as you're about to step onto the

Â walkway, you notice your shoelace is loose.

Â And the last thing is you want to do is get your shoelace caught in the mechanism

Â of the walkway. So you've got a choice.

Â You either stop and fasten your shoelace just before you step on the walkway, or

Â you step on the walkway and then stop to fasten your shoelace whilst the walkway is

Â moving you along. And there's no other options.

Â Those are the two options you have. Which option will get you to the gate

Â fastest? Now there are 2 ways you could try to go

Â about this. You could try to argue in terms of speed,

Â adding the speeds together and so forth, and if you do that you're going to run

Â into a problem. There's going to be some information that

Â you need that you don't have. For example, I haven't told you how long

Â the walkway is and how long have to walk in general.

Â So there's, there's, there's a problem if you try to use stunted methods.

Â However, you do have enough information to solve the problem.

Â But if you try to apply relative speed to the kind of methods you were taught to use

Â in, in high school, for solving problems about speed, relative speed, and so forth

Â you're going to run into problems, and you don't need that.

Â If you think about the problem, you should be able to solve it.

Â Well, I'm just going to leave you with that one.

Â It's an interesting problem, I think it's a rather neat problem.

Â There are many problems like this that, that you can really can just solve by

Â mathematical thinking. And the point is, this again is all about

Â this important notion Mathematical thinking.

Â It's powerful, if you can master mathematical thinking many problems can be

Â resolved without going into the complexity of applying mathematical techniques.

Â Mathematical techniques are what you need when mathematical thinking alone, doesn't

Â work. But often you can get by with just

Â mathematical thinking so, good luck on that one and enjoy it it's a lot of fun.

Â