So before we begin discussing probability, we need some very basic mathematics.
Now everyone listening to this lecture will have had set
notation at some point in their life and covered it from a very basic or
even more advanced mathematical perspective.
In probability, the set notation has the same rules,
of course, it's just a subset of ordinary set notation.
However, the interpretations of set notation are slightly different.
So usually when you talk about set notation, you talk about sum.
Uber space that contains everything.
Well, in statistics, we call this the sample space and
that we usually denote with an upper case omega and
this is the collection of all possible outcomes of an experiment.
So as in a simple example, let's conduct an experiment.
We roll a die, so the possible set of outcomes are one, two,
three, four, five or six.
Where here, we're not gonna play the sort of mental games that the die could land on
an edge or a corner or something like that.
It has to roll on one of the numbers and
then the sample space would be the integers from one to six.
An event is any subset of the sample space.
So for example, you could have the event that the die rolls even.
I that E is the set containing the numbers two, four and six.
Certain kinds of events are so
commonly talked about that we give them a separate name.
An elementary or simple event is the particular result of an experiment.
So for example, if the die roll is a four,
then usually we denote this for the lowercase omega.
Omega = 4.
Here, we don't tend to split hairs about the actual number four and
the set containing the element four.
But I think in the traditional definition, a simple event is the actual element for
not the set containing the element four.
But here, I don't think for our purposes that distinction will be necessary and
then it's always useful to talk about nothing.
So the null event is actually the event that nothing occurs.
The null event or empty set and
that's usually denoted with a letter here, which I'll just call null.
Again, the sets in probability theory follow all the same rules
as ordinary set notation.
Of course, because it is exactly ordinary set notation,
but just with different interpretations.
So when we say that an elementary event is an element of an event,
then that implies that E occurs when W occurs.
So for example, just looking back at the previous slide.
If our elementary event is that the die roll is a four and
the event is that it is even.
If you roll a four, then the roll is even.
If the elementary event is not in an event,
that implies that E does not occur when W occurs.
For example, if the elementary event is a five.
Five not being in the set of even number means that when you roll a five,
you have not rolled an even number.
We can follow along this logic.
So for example, E being a subset of F implies that the occurrence
of E implies the occurrence of F.
So for example, let's take E as the event that the die roll is even,
E equals 2, 4, 6 and F is the event that the die is either even or a 5.
Hence, F is the event two, four, six and five.
So two, four, five, six,
then the occurrence of E implies the occurrence of F.
That is if you roll an even die roll, then you have also
rolled in an element of the set of even die rolls plus five.
If the standard set intersection E intersect F
implies that both E and F occur.
So to give you a specific example of this, imagine that E is the event that
the die roll is even, F is the event that the die roll is a prime number.
So let's think of what the prime numbers would be on a die roll, that would be two,
three and five.
So E intersect F means that the die roll is both even and
prime that would just be the number two.
So the event E intersect F occurs means that you get both a even number and
prime number, which of course, in this case, would mean that you get a two.
E union F is the standard set notation for union, but
in probabilistic interpretation it means that at least one of E or F occur.
So in my previous example, it would mean that I either get
an even number or a prime number or both in the case of two.
If E intersects F is the null set, that means that both E and
F cannot simultaneously occur.
So imagine E is the set of even numbers, F is the set of odd numbers,
then you cannot roll a die that is both even and odd.
So, E intersect F will be the null set and
that's important enough that we give it its own name.
So in bold here, you see its own name.
That's called mutually exclusive.
So if we say that two events are mutually exclusive,
that means that they both cannot occur and
you frequently hear people use the phrase mutually exclusive incorrectly.
So what it technically means,
things are mutually exclusive if they cannot both simultaneously occur.
