It is about informal questions.
How can the past influence the future?
But okay, it's very philosophical.
Let me be more clear.
So, imagine that you have a dice and you roll it twice,
and you are asked,
"How probable is six?"
Imagine you have seen six in the first experiment and that's it.
This observation doesn't increase, decrease,
or doesn't leaves unchanged the probability of getting six in the second experiment.
So, in our mathematical setting,
we can compute it easily.
So, here, the space is just a pair of x and y.
Inside x is 1/6 and y is also between one and six,
and all combinations have the same probability.
And so we can compute the conditional probability of y equals to six,
under the condition that x is equal to six,
and without the conditions,
and it's easy to do this and both are 1/6.
There are 36 equiprobable outcomes.
Here, we have only one outcome.
This was as a fraction.
So here, we have only one outcome out of 36,
and here we have six outcome for all possible values of y.
So, we get 1/6.
This is also 6/36 because there are six possible value of x.
So, in our model,
the condition doesn't change the probability.
But model is model,
and real world is real world.
So, imagine there is a movie,
Rosencrantz and Guildenstern Are Dead,
I think it's called.
This movie starts with a long scene when
one of these two guys make a coin tossing at many times.
It's always head or maybe tail, but just the same result
many times and he become worried while the other guy ignores this problem.
But anyway, imagine you watch this movie and you see that it's head,
head, head, head, and so on.
So, the models still says that after you see 999 heads,
the probability of the next head is just 1/2 because you can
compute the probability to have 1,000 heads and
this probability to have 999 heads at something.
So, it's still 1/2.
But in the movie, probably you will have different feeling.
So, why you have this feeling?
Why you disagree with the model?
Probably, you will disagree.
The reason why you
think that the head looks more probable is not because in the model, it's more probable.
It's because you've been starting to think that the model doesn't fit the experiment.
So, there is probability theory and there is statistics.
So, probability theory somehow studies what are consequences of a given model.
Statistics tries to find a good model for experimental data.
So, there is a kind of separation between them.
So, in this case, a statistician would say,
"The model of fair coin is not a good model.
You should look for a better one and the better one is just the coin has two heads."
I don't remember whether in the movie somebody tried to check. Probably not.
It is a bit strange. So, a statistician will try to find a better model.
But if we assume that if we don't touch the model, if we assume this model,
in this model, we have still the probability 1/2.
Now, there is a very strange story about an American writer,
probably you know him from 19th century.
He's famous author of short stories.
I think he was one of the first authors of detective stories, and also,
he's well known for a poem about a raven which always says,
"Never more," independently of what you ask this raven.
It seems, at least,
if we take seriously what he writes in one of the stories,
it seemed that he believed that if you see some probability of six,
if you see six already,
it decreases the probability to get it another time.
Not increases as in our story about coin.
But he believed in the opposite direction and decrease the probability.
What is even more strange,
maybe it's kind of mystification. I don't know.
Maybe just a joke.
But in the story, it seems quite serious.
I will show you the quote.
But it seems that he believed that this decrease is just what probability theory says.
He complained that the general readers,
as he say this,
do not understand this,
a nice probability theory.
So, everything was mixed in his explanations.
So, the correct understanding was considered something what general readers believe,
because they are not aware of mathematical theory.
Some wrong thing was claimed to be a result of mathematical theory.
It looks very strange.
But it also reminds a joke
about a statistician who computed the probability to have two bombs in a plane,
and decide this probability is extremely small that it can be ignored.
He decided that you can bring one bomb yourself and then just keep it.
Then probability to have another bomb and to be in danger is extremely small.
But of course, it's a wrong reasoning,
but it's somehow similar to what Mr. Poe thinks.
But now, I don't want you to believe me,
I just want to show you the quote.
So, this is for a detective story.
Here is this statement,
the fact if you see already twice the six then says Mr. Poe,
it's a reason to believe the third six will not appear.
So, if you see two sixes,
it's the reason to bet against the third six.
Strange. Moreover, he claims,
it's difficult to convince the general readers, and indeed.
He explains why the general readers do not believe in this.
He says that if you make these dice, two sixes,
they are in the past, so to say,
and then you make a new experiment and
why the previous experiment can influence the new one.
So, this is explained very convincingly.
But strangely, Poe believes that this explanation is an error.
Of course, if he cannot explain, I don't know.
He doesn't want to explain it and he just say that,
look, the theory says that there is an error.
I have no space to explain it.
Often when people say things like this,
is just because they cannot explain it.
Anyway, everything is mixed here.
I don't know what the serious scholar thinks about this.
Was it a joke or did he really believe in this anyway?
But I just found this while reading the novel and it's funny and so I showed it to you,
but I didn't know what really this all should mean.
Alexander S. KulikovVisiting Professor
Vladimir PodolskiiAssociate Professor
