So we have three convincing arguments.

Now, we want to explain the correct answer.

Or if we are in a post-modern world,

it's not politically correct to say what is the correct answer,

we should say this, our position about

this problem and other people have different positions.

We don't make any claim, but if we want to be serious,

I will try to explain the correct answer and also explain why

the initial short setting is not good.

So, what do we think about this problem?

First, if the short setting is just say the short story,

then we cannot say what are the best strategy because the situation is not clear.

And in our elaborated setting the correct solution is the third one.

So indeed, changing the door increases the factor of winning chances by a factor of two.

So, why is it all these?

And there is some general advice how to deal with paradoxes in probability theory.

You should think about a repetitive experiment.

So, if there is no repetitive experiment,

it's not clear what is this experiment,

then the question is not well-posed.

But, if there is a repetitive experiment,

then you imagine it,

things become more clear.

So, in our case there is no problem in repetitions.

You can make this show every day,

and can make the random choices every day.

And then, you should imagine that the guess pull off the first strategy,

the second strategy, or the third strategy, and look on which percentage

of cases the guests will be.

So let's look at this.

So, by assumption, our prize is behind the door one,

door two, and door three,

in one third of all cases.

And in each of the third cases,

the guest makes independent guess and

so the guest's guess is correct with probability also one-third.

So, this is enough to say that the 'keep' strategy will win in one-third of the case.

If a guess always follows the 'keep' strategy,

she will win in one-third of the cases.

But in other third of the case, she will lose.

But if she would use the 'change' strategy,

then she would win exactly in the other cases.

In fact, this event can be considered as events of

the same probability space because we just have an experiment in one part,

for one set of the experiment.

The rest is not important but still that we have the same setting,

and there are cases when the 'keep' strategy is [inaudible] or the change strategy is given,

there are complimentary cases.

So the change is just the sure to win,

the probability two over three.

And we can also say that a random choice strategy will win

in half of the cases because

a new random choice we'll be correct in half of the cases by,

by definition of independent random choice.

So now we know the probabilities for all three strategies and

see that the third strategy is the best one.

And now, let me explain why that short

setting as the story below the picture in Wikipedia is not enough,

why you should not be hear the story.

I'll repeat it.

In search of a new car,

the player picks a door,

say door number one.

The game host then opens one of the other doors,

say door number three,

and there is a goat.

And then the player

has a chance to change the guess and pick door two instead of door one.

So this doesn't tell you what is the experiment,

what is a repetitive experiment.

This tells you only what happens once.

And this information is just not enough.

So this is a sequence of events in one case.

And, to see why it's not enough,

imagine that host of the show probably,

in advance knows, has some instructions what to do here,

maybe they are given or whatever.

But imagine the instructions like this.

So first, you let the guest make a guess.

And if the guess is incorrect,

then you open the door immediately and guess,

you save the money.

And even if the first guess is correct,

you still want to save the money,

you want to somehow confuse the guest.

And so you say, okay,

I open this door and now you can change.

Maybe the guess will change and then you'll

save your money because the first guess was correct.

So imagine that the host followed this protocol.

And in this case, it's obvious that,

if the host suggested to change the door,

it's a clear indication that the guess was correct,

and you should keep it, of course.

In this setting, the keep strategy will win the probability one in the case,

as one is applicable and you should always follow it.

So you see that

this story is consistent with

both descriptions with this one and our original description.

So we cannot distinguish between what is happening behind the scene,

and the recommendations depend on what is happening.

So this short description is not enough to make an informed decision about best strategy.