We have already seen an example in which we
have two events that are independent
but become dependent in a conditional model.
So that [independence] and conditional independence
is not the same.
We will now see another example in which a similar situation is
obtained.
The example is as follows.
We have two possible coins, coin A and coin B.
This is the model of the world given
that coin A has been chosen.
So this is a conditional model given
that we have in our hands coin A.
In this conditional model, the probability of heads is 0.9.
And, moreover, the probability of heads
is 0.9 in the second toss no matter
what happened in the first toss and so on as we continue.
So given a particular coin, we assume
that we have independent tosses.
This is another way of saying that we're
assuming conditional independence.
Within this conditional model, coin flips are independent.
And the same assumption is made in
the other possible conditional universe.
This is a universe in which we're dealing with coin B.
Once more, we have, conditionally
independent tosses.
And this time, the probability of heads at each toss is 0.1.
Suppose now that we choose one of the two coins.
Each coin is chosen with the same probability, 0.5.
So we're equally likely to obtain this coin--
and then start flipping it over and over--
or that coin-- and start flipping it over and over.
The question we will try to answer
is whether the coin tosses are independent.
And by this, we mean a question that
refers to the overall model.
In this general model, are the different coin tosses
independent?
Where you do not know ahead of time which coin is going to be.
We can approach this question by trying
to compare conditional and unconditional probabilities.
That's what independence is about.
Independence is about certain conditional probabilities
being the same as the unconditional probabilities.
So this here, this comparison here
is essentially the question of whether the 11th coin
toss is dependent or independent from what
happened in the first 10 coin tosses.
Let us calculate these probabilities.
For this one, we use the total probability theorem.
There's a certain probability that we have coin A,
and then we have the probability of heads in the 11th toss given
that it was coin A. There's also a certain probablility
that it's coin B and then a conditional probability
that we obtain heads given that it was coin B.
We use the numbers that are given in this example.
We have 0.5 probability of obtaining a particular coin,
0.9 probability of heads for coin A, 0.5 probability
that it's coin B, and 0.1 probability of heads
if it is indeed coin B.
We do the arithmetic, and we find
that the answer is 0.5, which makes perfect sense.
We have coins with different biases,
but the average bias is 0.5.
If we do not know which coin it's going to be,
the average bias is going to be 0.5.
So the probability of heads in any particular toss
is 0.5 if we do not know which coin it is.
Suppose now that someone told you
that the first 10 tosses were heads.
Will this affect your beliefs about what's
going to happen in the 11th toss?
We can calculate this quantity using
the definition of conditional probabilities,
or the Bayes' rule, but let us instead think intuitively.
If it is coin B, the events of 10 heads in a row
is extremely unlikely.
So if I see 10 heads in a row, then
I should conclude that there is almost certainty that I'm
dealing with coin A.
So the information that I'm given
tells me that I'm extremely likely to be dealing with coin
A. So we might as well condition on this equivalent information
that it is coin A that I'm dealing with.
But if it is coin A, then the probability of heads
is going to be equal to 0.9.
So the conditional probability is
quite different from the unconditional probability.
And therefore, information on the first 10 tosses
affects my beliefs about what's going
to happen in the [11th] toss.
And therefore, we do not have independence
between the different tosses.