Okay, so independence is going to be an important tool because I can compute from two probabilities.

The probability of their conjunction.

Which really gets the all these tables down by one.

One dimension.

And down by one mid dimension means.

That you have one order of magnitude less.

You go from two to the.

Two to the ten to two to the nine, which will already half down.

It's not a constant sum and but.

It's exponentially better.

So.

Let's.

Think about.

The dentist again.

And so we have the following joint probability distribution.

Cavity and toothache is.

12 not cavity and toothache even more unlikely.

Not toothache and cavity and so on and all has to add up.

To one.

So we have that.

So we have that how do we compute.

P of cavity.

P of cavity is.

0.12 plus 0.1.

The other one.

Yes, of course, but it doesn't change.

But it's 0 to not over one by the way.

By the way.

Okay, good.

So we sum up.

The rows.

Right.

P of cavity and toothache and P of cavity and not toothache and that adds up to 0 2.

So how do you add up cavity or toothache?

Right.

Or.

You use de Morgan's law.

Right.

Have cavity or toothache is not not cavity or not toothache.

We know that.

So that actually gives you the complement and so on.

And in a way that's really what we're doing here.

We use de Morgan's law.

And.

We get those things in.

We get.

We get.

This here.

And of course.

We know that there's a toothache.