Let us look at another example to bring this into context.
We consider the following setting.
This has got a classical advantage.
It is called Polya's urn scheme.
At heart, this is a, an urn and ball problem.
Imagine that one has a town, with r plus s residents.
r of the residents are royalists.
They support the king.
s of the residents are seditionists.
They want to implement insurrection, and replace the king.
Okay.
Not very fanciful, but colorful.
Imagine that the town's population grows steadily, if erratically.
At various epochs in time, groups of settlers arrive, a settlers at a time.
Imagine that you have buses pulling up.
Filled to capacity with a people in them.
And buses come at various times.
And each time a bus comes in, you add a people to the population of a town.
Now, the first individual that this bus load,
these arrivals meet, is either a royalist, or a seditionist?
If they meet a royalist first, then the entire group decides to don
royalist colors, and they become royalists.
If they meet a seditionist first,
then the entire group decides to join the rebellion and they become seditionists.
Now, here's a simple, very simple model, and of course,
is a very crude model of growth into two categories.
Now, if you take the fanciful trappings of this problem apart it's hard
to realize that what we are really postulating here is a very crude model.
In epidemiology, a crude model of the growth or spread of contagion.
Of course, we understand that in today's world, various
epidemics cause great consternation in various parts of the world.
Some of them lethal and deadly.
And so, it is of importance for, in public policy to understand or
to get some understanding of how infection spreads.
Here now, is a very simple model.
The idea here is that you can imagine a population of infected people,
and people who are well and perhaps better.
People who have, let's say oh, patches which resist infection.
Arrivals come in, and if they meet a person who is sick, then they fall sick.
If they meet a person who is inoculated and
has patches of resistance, then they develop resistance to the disease.
You understand?
It's a very simple and crude model of how disease might spread.
And how it has sequestered, how rapidly does the disease spread.
Now, if we abstract how the colorful language and
the verbiage around this, at heart, we have a ball and urn problem.
Imagine that the individuals represent balls.
Initially, we have r plus s balls in an urn.
r balls are red, s balls are black.
Each arrival corresponds to drawing a ball at random,
looking at its color, putting it back in the urn.
In other words, it's sampling with replacement.
And then, if the ball drawn is red, add a red ball to the urn.
If the ball drawn is black, add a black ball to the urn.
You understand, that this urn scheme is formally indistinguishable from this
fanciful story of royalists and seditionists, of illness spreading.
Okay.
So let's just focus on the essence here, on his ball and urn problem.
Now, our question is this, if we know that
the second and third groups become royalists.
If we know that the second and third inhabitants met by arrivals are red.
What are the chance that the first arrival,
the first group met are royalists?
In the ball and urn model, given that the second and
third balls drawn are red, what is the chance that the first ball drawn was red?
Now, before we launch into this,
it is always wise when dealing with problems to think about the setting.
Doodle, sketch, draw.
Try to get a right understanding of the circumstances around the problem.
And this particular problem is ripe for a graphical view