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1.1 The Monty Hall Problem

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1.1 The Monty Hall Problem

Probability and Statistics: To p or not to p?

Лондонский университет

We live in an uncertain and complex world, yet we continually have to make decisions in the present with uncertain future outcomes. Indeed, we should be on the look-out for "black swans" - low-probability high-impact events. To study, or not to study? To invest, or not to invest? To marry, or not to marry? While uncertainty makes decision-making difficult, it does at least make life exciting! If the entire future was known in advance, there would never be an element of surprise. Whether a good future or a bad future, it would be a known future. In this course we consider many useful tools to deal with uncertainty and help us to make informed (and hence better) decisions - essential skills for a lifetime of good decision-making. Key topics include quantifying uncertainty with probability, descriptive statistics, point and interval estimation of means and proportions, the basics of hypothesis testing, and a selection of multivariate applications of key terms and concepts seen throughout the course.

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Преподаватели

Dr James Abdey
Assistant Professorial Lecturer, LSE

Текст видео

[MUSIC]

Are you feeling lucky?

Well, I want us to play a little game together.

Imagine there are three doors A, B, and C.

Now we're going to play this game collectively,

although this is by a recorded video.

But I'm going to explain the rules of the game first of all, and

then we're going to play it.

Now behind one of these doors is a star price.

Let's imagine it's a brand new sports car.

Behind the other two doors are some sort of booby prize.

Let's say, there are two goats behind the other two doors.

Now I'm the game show host, and I know where the sports car is.

And it's behind one of those three doors, and

it will stay behind one of those three doors.

Now it's up to you to choose one of these three doors.

I'll assume your choice in a moment, but

whichever door you choose, I will not open.

I will open one of the other doors and reveal what is behind that.

And then, I will throw the decision back to you when you may either stick with your

original choice of door or to switch to the other unopened door.

So to be clear, I know where the sports car is and

of course where the goats are, but you don't.

The sports car will remain fixed behind one of those three doors.

So you'll make a decision.

I will not open the door you choose.

I will open one of the remaining two.

And then, the decision is thrown back to you.

Now ordinarily, I would teach this in a class and put it to the democratic vote.

So if I ask you to choose one of these doors A, B, and C.

Well, given this is a recording, let's assumed you've chosen door A.

Now frankly, this game works whichever door you originally chose.

So door A has been selected by you.

And as I said, I would not open the door which you chose.

So door A will remain firmly closed.

I am going to open one of the other two doors.

And let's imagine I choose to open door B and I reveal a goat.

So now you know what is behind door B.

Originally, it was just an unopened door and you had no idea what was behind it.

So I said having opened one of the other doors, I would return the decision to you,

which means you could either stick with your original choice, door A in this case,

or you may change your mind and switch to the other unopened door, door C.

So just by yourself,

just make your own judgement about which one you're going to choose.

Okay, now of course I don't know which one you've decided.

But before we perhaps conclude the game, I'd liked you to consider this

as a problem of decision making under uncertainly.

So let's revisit this probabilistically.

So consider the start of the game.

Doors A, B, C, all unopened,

you had a one-third probability that the sports car is behind each of them.

So in my imaginary world you chose door A initially and

I said, I would not open the door that you choose.

So the fact that door A remains unopened is of no surprise to you, and

indeed gives you no new information.

So if nothing changes about the world,

if you receive no new information about something, there should be no revision or

updating of your beliefs because there is no new information to absorb.

So that initial probability of a third attached to door A, remains a third.

Turning to door B, initially was unopened and

you had a one-third chance of the sports car being behind it.

My opening of door B, revealing the goat did give you some new information.

And indeed now you know what is behind door B.

So armed with that new data,

that new information, you were able to revise or update this probability.

In this case, the revision is downwards.

It was originally a third, and now becomes a probability of 0,

an impossible event because we know for a fact the goat is behind there.

So now let's turn to door C.

So if I ask you the question, did you learn anything about door C?

Many people would probably answer no,

because they see the action of door B being opened.

However, you have to be conscious that I have the option of opening door C.

It was only door A, which had to remain closed.

So in fact, your seeing me reveal the goat behind door B, simultaneously,

you are actually observing me consciously choosing not to open door C.

So in fact, you do learn something about door C.

So let's consider, the two possible explanations for

why I opened door B in the first place.

Let's imagine the sports car was behind door A, you choose it and

therefore there are goats behind doors B and C.

I can't open door A and therefore, which ever door B or

C I open, I would be revealing a goat.

And in this world therefore,

I am indifferent between opening door B or door C.

So that is one possible explanation.

Of course, the other explanation is that the sports car is behind door C.

Your choosing of door A prevents me from opening it, clearly I'm not going to show

you where the sports car is initially, so I'm not going to be opening door C.

And it's the only option available to me is to open door B.

Now as I know where the sports car is,

I know which of those two explanations is the correct one.

Of course though, you do not.

So you know that there's a possibility that I was forced to open door B,

and that original probability of a third that you assigned to door B

is in fact transferred and inherited to door C.

And hence, its revised probability

of having the sports car behind it is now two-thirds.

So the original probability distribution of a third, a third, and

a third is now revise to a third, zero, and two-thirds.

Now I'm a nice guy, so knowing this new information,

let me return the decision back to you.

These are now your revised probabilities of winning the sports car.

So I'm guessing none of you is going to choose door B.

So think about whether you would choose door A or door C.

Now I'm going to imagine you are going to play to the probabilities here.

And a rational person would clearly choose the option with a greater chance of

success.

So I'm imagining you have chosen now door C.

Well, let's reveal what's behind door C.

And what do we find?

No, it's a goat.

So did you make the right decision?

Well, of course,

with the benefit of hindsight, now you know exactly what is where.

Let's just reveal the sports car behind door A.

Now those revised probabilities are 1, 0, and

0 with certainty, you now know where the sports car was.

But did you make the right decision by choosing door C?

Absolutely, based on the uncertainty that you faced it's quite

rational to play to the probabilities.

But James, I did the right thing according to you.

Yet, I've ended up with the bad outcome.

True, in this one instance, yes you have.

Of course, one has to consider here the element of luck.

And luck comes in two types, good luck and bad luck.

Now in this instance, you experienced some bad luck.

You did the right thing and yet, on this occasion it wasn't the desirable outcome.

And had you stopped with your original choice of door A,

you would now be the owner of brand new sports car.

But that's not to dismiss this concept of probability and

doing what is optimal based on the available information.

In that, imagine you played this game a very large number of times.

By playing to the probabilities, by taking that strategy,

the optimal strategy of switching door each time, then on average, two-thirds

of the time you would win versus one-third of the time you would lose.

So going forward, and

we'll be saying a lot more about quantifying probabilities, and

we will revisit this particular game more formally using some probability theory.

However, I don't promise you that you're going to be making the right decision

every single time.

We can all be unlucky,

sometimes things don't turn out as we might have wished they have done.

However, in the long run, if you play through the probabilities,

you're going to win far more often than you lose.

So this is a perhaps a paradox over a problem, people very often assume,

it's a 50, 50 change of getting a sports car after the initial goat is revealed.

However, probability can often be very counter-intuitive.

So this is just a little taste of decision making under uncertainty.

You need to perhaps clear some of this probabilistic fog,

as we shall do later on in the course.

[MUSIC]

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