Okay, so now probability.

Any, any event has a probability of happening, 'kay?

So let me give you an example.

If you wake up at a random hour of the day.

So you just wake at some random hour of the day.

Say you have jet lag or something.

And I ask you what is a property that the event at the time is between 10 a.m.,.

. and 11 a.m.

Let's try to calculate this.

Well, there are 24 hours in a day.

So, you have a set that consists of 24 elements, one for each hour.

12 a.m.,

. 1 a.m.,.

. 2 a.m.

All the way to 10 a.m.,

. 11 a.m.,.

. 11 p.m.

Okay, so when I say, and I limit like 1 a.m.,

. I mean the hour between 1 a.m.,.

. and 1:59 a.m.

Similarly, 10 a.m.

Means between 10 a.m.

And 10:59 a.m.

Out of the set upgrade for elements, you pick one at random.

And the probability that you're going to pick 10 a.m.

Is essentially 1 divided by 24, because you're picking one element at random.

And there are 24 elements in the set.

And the probability that you pick that one special element 10 a.m.

Is one over 24 because you, you want to pick, you want to be lucky.

Of that, to pick that ten.

So the probability that if you wake up at a random hour of the day and

the time is between 10 a.m.

And 11 a.m.

Then that probability is 1 over 24.

Now there will be multiple events and you might need to calculate the probability of

conjunctions or disjunctions of these events.

I'll describe what these are soon.

So say E1 is one event and E2 is another event.

And E1 and E2 are independent of each other.

This means that E1 does not influence E2, E2 does not influence E1.

Then the probability that E1 and

E2 both happen is essentially the multiplication of these probabilities.

So you take the probability of E1.

Multiply it by the probability of E2, and

that gives you the probability that both E1 and E2 are true.

Let's discuss an example of this.

Suppose you have three shirts.

They're colored blue, green and red.

And also you wake up at a random hour and blindly pick a shirt.

You put your hand into your closet and you blindly pick out a shirt.

What is the probability that you woke up between 10 a.m.

And 11 a.m.

And that you picked a green shirt to wear?

Well the probability that you woke up between 10 and

11 is essentially 1 over 24.

Like we calculated on the previous slide.

The probability that you picked the green shirt is essentially one out of three

because there are three colors of shirts.

You have three shirts.

And you want to pick the green one.

You want to be lucky to build the green one so that's one over three.

So the probability that both these elements are true that you woke up between

10 and 11 and that you wore the green shirt,

is the multiplication of the probabilities of those events.

So it's 1 over 24 times 1 over 3.

And that gives you 1 over 72.

One of the things that you have to be careful about here is that if E1 and

E2 are not independent, meaning that they influence each other in some way.

Then you can not multiply the probabilities with each other.

Okay, so for instance if