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Hello and welcome to week two,

where we're going to be considering quantifying uncertainty with probability.

So in the first week, this was just really the introduction to the course, and

I wanted to get you thinking about the general concepts of

decision-making under uncertainty.

And for example, we kicked off with the Monty Hall problem.

But of course, now we need to formalize things somewhat more.

We need to decide what exactly is probability and how do we quantify it?

How do we determine this?

So to assist us with this, we do need to introduce some key vocabularies,

some key lexicon, if you will.

So we begin with the concept of an experiment, indeed a random experiment.

Now examples for this could be trivial things such as tossing a coin and

seeing which is the uppermost face.

Maybe it's rolling a die and seeing the score on the uppermost face there.

Or maybe it could be more sort of real world examples,

whereby we look at some stock index like the FTSE 100 and

see what the change was in the value of that index on a particular trading day.

So there's some random experiment,

which could lead to one of several possible outcomes.

Now sometimes the number of outcomes might be quite small.

Tossing the coin, we've only really got two options here,

either the uppermost face is going to be heads or it's going to be tails.

If we consider extending it to rolling a die, well of course there,

on a standard die, you have six possible values for that uppermost face.

The integers 1, 2, 3, 4, 5 and 6.

If instead we consider the FTSE 100 Index, well of course, there are lots of

possible values we can have for percentage change on a particular trading day.

So we have a random experiment, which results in a particular outcome.

But we need to appreciate that there are several possible outcomes.

So our third term to introduce is that of a sample space.

And this is effectively the set of all possible outcomes in an experiment.

So we're sort of now considering the language of what we call set theory.

So our notation, let's say, for a sample space would be the letter S.

And we're going to have these sort of curly braces to reflect some set.

And within these curly braces will be all of the elementary outcomes,

all of the members of that set.

So just to revisit the sort of three examples we've worked with thus far,

if we consider the experiment of tossing a coin, our sample space will

have as its members the two possible outcomes of heads and tails.

Which we may wish to denote, let's say, by H and T for heads and tails, respectively.

Of course, using intuitive notation is always a very sensible idea.

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So with that coin toss example, there we see the elementary outcomes of

the sample space are not numerical values, but in this case, letters.

H for heads and T for tails.

Of course, sometimes and very often, the outcomes of our sample

space of our experiment may well indeed be numerical values.

So if we consider that rolling a die, there our sample space would consist of

the elements 1, 2, 3, 4, 5, and 6, representing all possible values,

which could occur on the uppermost face of the die.

So there we have a larger sample spaces with six members, or six elements

within it, contrasted to just the two outcomes in the case of tossing a coin.

Now our third example was the percentage change in some stock index,

such as the FTSE 100.

So here we're going to have a larger sample space than in the coin toss and

the die example.

because let's consider,

what are the possible percentage changes in the FTSE 100 in a given trading day?

Well, in principle, a stock market could lose all of its value, the entire

market capitalization of every company on that index could be completely wiped off.

The largest possible percentage change would be a loss of 100% value.

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Now of course, an index can't lose more than all of its value, so

we are constrained there's a maximum loss of minus their 100%.

But if now we consider the upside, well, in principle there is no law

stopping a stock price from exceeding some threshold level.

So in principle, in a given trading day,

the stock market could achieve an infinite percentage return.

And hence, our sample space, the set of all possible outcomes for

this random experiment of the percentage change in the stock market

on a given trading day would run from a -100% up to infinity in principle.

Now clearly,

I can't recall a day when the entire market capitalization was wiped out.

If such an event occurred,

I'd imagine the world economy is now a very mad place indeed.

Nor can I recall a situation where stock markets have risen to an infinite level.

However, in principle,

that represents the set of all possible outcomes of our experiment.

So we have a random experiment resulting in various outcomes, and

the sample space is the set of all possible outcomes in that experiment.

Now of course, by construction,

the occurrence of the sample space must be a certain event.

We know if we toss a coin, it's either going to be heads or tails.

So it is certain that, that event itself, the sample space, will occur.

We know with certainty, if we roll a die, we must get an outcome of 1,

2, 3, 4, 5 or 6.

So it is certain that one of those six outcomes will

be the uppermost face on the die.

And similarly, in a given trading day, we know that whatever the percentage

change in that stock market is must be somewhere between -100, and

in principle, up to infinity.

So of course, those sample spaces, yes,

it's nice to clarify all possible outcomes in our experiment.

But of particular interest to us will be something that we call an event or a set.

And this is going to be a subset of our sample space.

So in the case of tossing a coin, maybe of interest to us is getting or

tossing heads.

Maybe at the casino if we're rolling the die,

we win a prize if let's say, we roll a 6.

Maybe as far as the stock market is concerned,

if we own some sort of a market tracking index, then perhaps we are concerned or

interested in the event where the stock market increases in value.

So we could denote, let's say, by the letter A,

some specific subset of the sample space which is of interest to us.

For example, the event A might be tossing heads, it may be rolling a 6,

it may be getting an increase in the value of the stock market index.

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So as far as probability is concerned,

we will define all probabilities to be some value over the unit interval.

That is the range of values spanning from 0 to 1, where if an event has

a probability of occurring of 0, this would indicate an impossible event.

So there's no chance of that occurring.

And at the other extreme, if an event has a probability of occurring of 1,

then this would indicate a certain event.

Now as we had sort of trivially suggested, the sample space,

given we define it as the set of all possible outcomes in this experiment,

this does represent a certain occurrence.

So we could actually introduce some of our first probabilities and

say that the probability of our sample space S is going to be equal to 1.

Of course though, what we're really interested in are those subsets.

Those particular events, which represent a subset of that sample space.

And it's trivial to know that I must get a 1, 2, 3, 4, 5 or 6 if I roll a die, but

I'm really interested in the probability of one of those specific outcomes,

eg, a 6.

So other than those extremes of an impossible event with a probability of 0,

a certain event with a probability of 1, what about all of these

possible events which are neither certain nor impossible events?

So how do we quantify, how do we assign a probability of some event

A such that it lies somewhere within that unit interval?

Well, of course, there are different ways which we can actually attach probabilities

to these different events.

Sometimes these probabilities may simply be determined subjectively.

Maybe it's done through a sort of relative frequency approach through

experimentation.

And thirdly, it may be done theoretically.

Well, as far as this small section is concerned,

we'll just briefly consider the first two of those, namely coming up

with probabilities subjectively and also through experimentation.

And we will consider the theoretical approach in at the next section.

So subjective probabilities, when will World War III occur?

Well, there's a happy thought.

Well, of course, none of us knows for sure.

There's uncertainty about the future.

Yet some people have had an attempt to try and

determine how close we are to some sort of Armageddon event.

So I'd encourage you to do a search for the Doomsday Clock.

Now this was created in 1947 by a load of scientists to

determine just how close to midnight the world is, ie,

how close is humanity to some sort of man-made catastrophic event?

Now when the doomsday clock was designed,

they were mainly concerned about the threat of a sort of global nuclear war.

Well, more recently, people have considered things such as climate

change as also being of a potentially existential issue for humanity.

Now of course, determining just how close we are to this catastrophic event is

a highly subjective thing.

What's the probability that World War III will break out next week, next month,

next year?

Well, of course, it's very hard to actually quantify the likelihood of this.

And clearly, different people will have different subjective estimates of this.

True, we can all look at the news, we can see what's going on geopolitically around

the world, and then make a judgment call.

But for example, I might attach a 0.1% chance to World War III next year.

You might attach, let's say, a 0.5% chance.

Who's right and who's wrong?

Well, we can't say definitively.

It's a subjective choice.

So sometimes for events, which we can't actually replicate in an experimental

sense, we typically have to resort to subjective estimates of probabilities.

Well, sometimes we can conduct an experiment.

Suppose I have a coin, maybe it's a fair coin, maybe it's not,

I don't know for sure.

But I could toss this coin a very large number of times and

see the frequency of a particular outcome.

So for example, if I toss this coin 10,000 times,

and let's say I observed 5,050 heads say,

then approximately 50% of the time a head occurred.

And using this sort of relative frequency approach to probability,

we might lay claim to say there's a 50% chance of heads and a 50% chance of tails.

IE, conduct an experiment a very large number of times and

the proportion of times a particular outcome occurs, we might say,

represents the probability of that particular event.

So those are our first two looks at how we may come up with numerical estimates of

probabilities.

In the next section,

we're going to consider things more from a theoretical perspective.

So we are now more formalizing our approach to probability.

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