So everybody who has a value of X = 1 has a propensity score, 0.1.

Now let's look at the other group of people,

those are individuals who have X = 0.

Here are the probabilities that A = 1 which is a probability of treatment,

given that X = 0 is equal to 0.8.

So in other words, the subpopulation of people who have X = 0,

their propensity score is 0.8 which means that 80% of

people who have X = 0 will receive the treatment.

So people with X = 1 will unlikely to receive the treatment

where is people with X = 0 are very likely to receive it.

We could depict this in a picture where we separate the treated and

controls via this vertical line, and

then we're using just blue, and red to indicate the values of X.

So, blue is for X = 1 and red is for X = 0.

And this is just a hypothetical kind of situation where, but

that's corresponding to what we saw with the propensity score where for

the people with X = 1, the large majority of them.

And in fact, 90% of them are in the control group.

And for X = 0, we see that 4 out of 5 of them are in the treatment group which

again, corresponds to what we saw for our assumption about the propensity score.

So, let's just focus on one of these groups for now.

So, this is the X = 1 group.

So, this is a subpopulation.

And in that subpopulation, for everyone one treated subject,

you would expect to have nine control subjects.

Or in other words, one out of every ten people with X = 1 is treated.

So, that's what we'd expect on average.

So out of 10 total people with X = 1, we'd expect for one of them to be treated.

Now, let's imagine we're going to do propensity score matching.

So you see we have this imbalance here in the sense that among people with X = 1,

they are way more likely to be in the control group.

Whereas if you had a randomized trial and you were strictly randomizing people

to treat it in control groups, you would expect an equal number of treated and

controlled subjects in the subpopulation.

So, what propensity score matching would do then is remember that.

Because there's only one covariant here,

everybody with that is equal to 1 has the same value of a propensity score.

So all of these ten blue dots here, they all have the same propensity score.

So if we were going to do propensity score matching, what we would do is we would

match one treated subject to one here randomly selected control subject.

So with propensity score matching,

the data that we would end up using is right here.

We'd just use all of that and

what we see then is that there's one person in the treatment group now.

There's also one person in the control group, but

there was originally one person in the treated group.

So this one person in the treated group.