WARNING: "This is written by an entirely mad CHaOS committee member a bit to close to Christmas. Please do the experiment however you want, and don't feel you need to imitate a goat or the style this is written in. This summary will give you some idea what is going on... A wonderful experiment of probability involving a cuddly goat called Bertrand, a couple of coins, three doors, and three boxes. It involves tossing a fair coin, and seeing how many heads you get in a row before getting a tail (and the probability distribution you get from that). It explains the Monty Hall Problem, and the importance of prior probabilities (three doors, with the prize behind one door). There are then more complex problems, involving three boxes, each with two coins inside them!"

Monty Hall’s Goat (Bertrand)

Hello boys and girls. I said HELLO boys and girls

* Hello Bertrand *

That wasn’t very loud now was it – you can do better than that. HELLO BOYS AND GIRLS

* Hello Bertrand *

Deary me, you still seem to be hung over from the new year. I said ** HELLO BOYS AND GIRLS **

* HELLO BERTRAND *

Oh dear, oh dear, no need to shout. Honestly what are the young like these days. I think I should tell you a little bit about me before I got here. I used to have a job as a pantomime horse, but quit while I was a head.

So what is probability? Well it is the chance of something happening to you. What is the probability that I am going to give you £50000? Given that I am a goat with no worldly possessions, 0. What is the probability that I am going to be sitting on this table in 2 seconds time – 100%. We can measure probability from a scale of 0-1, or multiply it by 100 to get

Right, now where did I put down my coin

* It’s behind you *

Oh no it isn’t

* Oh yes it is *

Oh NO IT ISN’T

* Oh yes it is! *

Oh yes, so it is. I thought you were just telling me my career was behind me… Anyway, who wants a fun game where I make up the rule. How many times do I have to toss this coin before it I get a tail? Hands up for once. Hands up for twice. Anyone thinking I am going to get three heads in a row before getting a tail? Anyone more than that? What is the best one to go for?

Well if you think about it, you have a 50% chance that it is going to be the first one. Then a 25% chance that I will get precisely one head followed by one tail, and then 12.5% two heads and then a tail. Three heads 6.25%, four heads 3.125% etc… We can draw this in a tree diagram, and get some stick for this… Perhaps we ought to turn a new leaf… OK, are you ready for this… Drumroll please – no no no – I mean hit the drum with a stick in a rapid way, not roll the drums down the hill and of a cliff into water (bom bom tish). Right ready for this, here we go….

[Bertrand’s coin has two heads]

We have a head. Bad luck to those who said that you would get zero heads. Now let us see if the one headed people were correct

[Second toss]

No head again. What about two heads?

[Third toss]

Still a no… Hmm that is unfortunate. Is Three a magic number?

[Fourth Toss]

No. Hmmm, let’s keep going

[Fifth, Sixth, Seventh Toss, till]

* You are cheating – there is no tails on this coin *

Oh No I am not

* Oh Yes you are *

Oh no I am not

* Oh yes you are *

Oh no I am not

* Oh Yes you are *

Oh well maybe you are right. Fine fine, spoil all my fun. Let’s use an actual coin – here you can check it does have a head and tail. Let’s note down on a piece of paper what it will look like – remember it is the number of heads we get before getting a tail…

[Paper should look like

0 – iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii

1 – iiiiiiiiiiiiiiiiiiiiiiiii

2 – iiiiiiiiiiiii

3 – iiiiiii

4 – iiiii

5 – ii

6 – i

7 –

8 – i

9 –

]

Right, well you kids (in America) have had the time of your life tossing all those coins – let us see if we can take tree diagrams just that little bit further… This is now a Mony Hall Problem. Here is my house, which has three doors. I am going to hide behind one of them… So which one am I behind?

[Kid picks one]

Ah wrong one. But how much chance did you have? Draw out a tree digram – 1 in 3 chance, right? How much chance do you have of getting it right both times? Less right? Compared to not getting it right either time, a lot less, right? So then, let us place a variation, where this time I use my beautiful assistant

[Looks at demonstrator]

ARGH! You were not my beautiful assistant from last week. Where has he gone? What I am stuck with you? But you don’t even have a beard… Oh well, needs must I suppose, but do try and grow a beard for next time.

Right, so like last time, I am hiding behind one of the doors of my house [the one with three doors where they have no idea which one you put Bertrand (the goat behind)]. Pick one, but this time you don’t get to open it yet. Beautiful assistant, open one of the doors I am not behind. Right here is the tricky question – you now have a chance to swap doors from your original choice – should you swap or not? Does it make a difference?

OOoooooohhhhhhhhh – that’s more like it – audience participation and all that jazz. Take your time, think carefully. Does it matter??? Drum roll please NNNNOOOOOOOOO not another one down a hill and off a cliff (bom bom tish). Look, that is getting expensive. Anyway, so what is the answer. Surely it doesn’t matter?

Firstly, my name is not Shirley, it is Bertrand. Secondly, it does matter. You should always swap. Don’t believe me? Let’s do it a few times, and see which action would have won you me (because heck, I am better to win than the demonstrator standing next door to me).

Swap - iiiiiiiiiiiiii

Don’t Swap - iiiiiii

So why do you win it about double if you swap than if you don’t swap? Simples – You have a two thirds chance of getting the goat rather than a one third chance. And no, I haven’t suddenly become a well dressed meerkat. By the not-that-beautiful-but-still-helpful assistant (you can grow a beard one day, I have faith in you) removing the wrong choice out of those two, you get two choices for the price of one.

Right, my still-not-attractive assistant has got annoyed with me now, and has put on a pair of horns [put on devil horns]. In this one, I am hiding behind one of the three. You get to pick one door. Out of the other two doors, he will open one of the doors and take away whatever is behind it. He will always take away me if he can, else he picks randomly from the two doors. Should you swap, or not swap?

Swap -

Don’t sway - iiiiiiiiiiiiiiiiiiiiiii

You now only have a one third chance of getting me, and that is only if you picked me at first. If you didn’t pick me, then you cannot get me. If you did pick me, then I am protected. Thus you should never swap

Right, remove those horns, and put on a halo instead. This time my wonderful-yet-not-bearded assistant allows you to choose one of the three doors. He then looks at the other two doors, and opens a door where myself (Bertard the goat) is not standing. He then ONLY offers you the chance to switch from your current door if it is beneficial i.e. if I am standing behind the door that you haven’t currently selected. It is therefore always beneficial to switch

Right, remove that halo, and put on the Dunce’s cap instead. This time, my brilliant-yet-lacking-facial-hair-clearly-showing-lack-of-intelligence instead randomly picks one of the other two doors without looking. There are therefore the following outcomes

Dunce picks the goat – iiiiiiiiiiii

Don’t swap to win – iiiiiiiiiiii

Do swap to win – iiiiiiiiiiiii

Equal chance of all 3.

Thank you, thank you for the applause, (hint hint), but there is more, as I know you were all shouting for an Encore. Just before we do, do any of the rest of you want to grow facial hair? I always think a goatie is the pinnacle of fashion, being a goat myself…

There is more!!! I now have three boxes sitting on the table, or mini-suitcases of mine, in case I want to go on holiday somewhere. Now in one of them, box GG, there are 2 gold coins, in another, box SS, 2 silver coins, and in a third, box GS, one gold and one silver coin. I can thankfully tell the difference through feel – my optician recently told me I was colour blind. It was a real shot out of the orange. Hoorah! Hope you are not feeling all boxed in!!!

I want you to pick one random box – good that one. Which box is it – you have a 1 in 3 chance of guessing.

OK – to make it easier I will pick out one coin at random and tell you there is one gold coin – what is the chance of getting each box. OK – you are predicting 50:50 between box GG and box GS. That is incorrect. Let us see what actually happens

Pick G out first

Box GG – iiiiiiiiiiiiiiiiiiiiiiiii

Box GS – iiiiiiiiiiii

Box SS –

Pick S out first

Box GG –

Box GS – iiiiiiiiiiii

Box SS – iiiiiiiiiiiiiiiiiiiiiiiii

As you can see, if we combine both scenarios together, there is an even chance of getting any of the three boxes. But as if I pick out a gold coin it is twice as likely to be box GG rather than box GS, as box GG has two gold coins in it…

What, you want another encore? OK, here is the scenario, but before I do, did you know I have been married sixteen times! four richer, four poorer, four better, four worse. After the loss of each partner, I was down in the dumps and got a new hat. No wonder my hats always looked terrible.

Three CHaOS demonstrators, Alice, Bob and Charlie are sitting out the back. However, I want my house cleaned, and I need two demonstrators to do this. I have decided which two will need to do this, but haven’t told any of them yet. Alice comes to me, and begs me to tell her one person who is definitely cleaning my house. I tell her Bob is definitely cleaning my house. She then goes and tells Charlie, believing that she now has odds of 0.5 of getting off cleaning my house. But Charlie believes that he now has odds of 0.6666 of not doing cleaning – who is correct?

The answer is Charlie. Alice was guaranteed to be told someone who has to do cleaning, and so it hasn’t affect her chances of having to do the cleaning. However, Charlie didn’t, and so his chance of not doing the cleaning has gone up. Confused? Me too. Let’s put this into a table

Not cleaning |
Me: "B cleaning" |
warden: "C cleaning" |
sum |

Alice |
1/6 |
1/6 |
1/3 |

Bob |
0 |
1/3 |
1/3 |

Charlie |
1/3 |
0 |
1/3 |

So being told that Bob is cleaning means that Bob has a 0 % chance of getting off, and relatively Charlie has twice the chance than Alice of not cleaning… Bob was the most depressed about this news, and ate 15 litres of low-fat yoghurt – no wonder he was Mullered by the end of the day…

Right, if you want further discussion, go and look at Bayes Theorem

https://en.wikipedia.org/wiki/Bayes%27_theorem

That’s all folks! Now it’s time to make my exit before anyone makes me their escape goat.

Sources:

https://en.wikipedia.org/wiki/Monty_Hall_problem

https://en.wikipedia.org/wiki/Three_Prisoners_problem

https://en.wikipedia.org/wiki/Bertrand%27s_box_paradox

PS By the way, you have spent the last twenty minutes talking to a goat – I suggest you find your marbles once again!!!