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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