I made a big caveat about

prior probabilities being the no knowledge variant.

So we're going to now look at what about probabilities,

where you have knowledge.

The first thing you realize is

that probabilities actually change.

And since probabilities are about incomplete knowledge,

they change as our knowledge becomes more complete.

Very simple example.

The probability that you're going

to miss your connection in Nunback really

gets different when you see that sign.

The S-Bahn has a delay of 30 minutes.

You can be almost sure that you're

going to miss your connection.

The probability of a cavity changes

when you know that the person has a toothache.

You know new stuff, probabilities change.

And that's a very, very important thing.

It doesn't affect, of course, the prior probabilities.

They're kind of the starting point

where you start gathering knowledge,

and then they all changes.

And so what we're going to be more interested in really

is the probability of a given that we know b.

Remember our convention?

We have the random variables or propositions, capital A

and capital B, where little a and little b

are actually outcomes in their ranges.

So this is the probability, for instance, of cavity

as the outcome, cavity equals true,

under the knowledge that we have toothache being true.

So that's really the object we're interested in.

Here we are.

So how do you do that?

Well, actually, it's a very simple formula.

You look at the ratio of having cavity and toothache

divided by all the cases where there's a toothache already.

And so that's cavity if toothache.

Very simple, very effective.

So there's really a subset construction going on.

You're really interested in this over that.

What are the likelihood of things

where you have both cavity and toothache given the sample

set of toothache?

Those are the things we're interested in.

What we can do with prior probabilities,

we can do with conditional probabilities.

So again, all face p vectorizes everything.

And we can think of the conditional probability