Monty Hall, with the earthquake... and musings

...further to my last post...

So, when Monty opens the door, that does not alter the probability that the prize was in (b) - the two non-chosen doors. We know that regardless of whether the prize is in (a) or (b), he's not going to reveal the prize when he opens the door. When he opens the door what it does is guarantee that if it was in (b), it is now behind the one remaining (b) door, which now has the whole of the 2/3 probability to itself.

But, when the earthquake opens one of the (b) doors and shows the prize is not there, that does alter the (b) probability. The earthquake might have revealed the prize, but it didn't, so what it tells us is just that it's not behind that particular door. The prize is not behind that door but is equally likely to be behind either of the others. The (b) probability drops to 1/2 and the (a) probability increases to a half.

David Mackay has a much more rigorous explanation*. However, my point is that while I am now fully convinced that I 'understand' what's going on, my initial intuition has been twice overturned. For a while previously I was fully confident of the opposite opinion. So, how can I ever trust what I earnestly believe to be true? New insights might overturn what I believe. It's back to this idea of the provisional again. What we believe can only ever be provisional. That's the only knowledge on offer.

* If you want to read it, David Mackay has made his excellent book available for free download here. The problems are on page 57 and the solutions on page 61.

Monty Hall problem: with or without an earthquake

There's nothing new about this (I'd seen the basic problem years ago but only recently came across the variation with the earthquake, in Prof David Mackay's book on information theory), but I think it is fascinating - and makes you think about what you know and how you know it, as I shall explain.

1) The Monty Hall problem. A quiz show. There are three doors. Behind one of them is a valuable prize. The contestant chooses one door. The host (Monty), who knows where the prize is, then opens one of the other doors revealing nothing.

The contestant is then given the chance to change their mind about which door to choose - ie either keep the door originally chose or swap to the other one that wasn't opened.

What should they do - and does it make any difference?

I'll leave some white space so you can't see the answer straight away, in case you want to think about it...
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Answer 1. The answer is that the contestant should swap. For me - and I think for most people - this was really counter-intuitive when I first met it. However, after pondering it for a while, it switched from "no, I just don't believe it" to "yes, actually, it is pretty clear". The argument I used to understand it was as follows. After choosing a door, we now think about two possibilities:

(a) prize is behind the chosen door = win
(b) prize is behind one of the two not chosen = lose.

Since each door is equally likely to conceal the prize, the prob of (a) is 1/3 and the prob of (b) 2/3.

When Monty opens one of the doors this doesn't change these probabilities, but now possibility (b) applies only to one door. So if the contestant swaps they get a 2/3 prob of winning rather than 1/3.

It is more obvious if you imagine lots of doors - say 10. The prize is much more likely (9/10) to be behind one of the nine not chosen than the one chosen (1/10). If Monty then opens the eight others, it is probably behind the one he didn't open among the (b) doors, rather than one (a) door.

2. With the earthquake. Now, same as the original problem, but just before Monty opens a door, there is an earthquake that opens a door by chance. This happens to be one the doors that the contest did not choose, and the prize is not behind it. What should the contestant do now?
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This time it makes no difference whether the contestant swaps or not! It is the same probability - 1/2 - either way. Again, I couldn't believe this at first, but, now, what do know, it seems perfectly reasonable. But no time to explain now. Will come back at a later date. [In the next post]

Information theory and fine wine

In the June 2009 IEEE Information Society newsletter, originally in The World of Fine Wine 18, 2007, is:

Spirits of Place. An information-theoretic account of terroir and typicity.

Douglass Smith

I'm not sure how seriously we are supposed to take it, or how tic (tongue in cheek) it is. My initial reaction was to look for '1st April' somewhere, but on reflection it is entirely in the line with what I was doing with art previously, and that was entirely serious.

The author explores the idea of a communication channel, communicating the terroir to the drinker.  He models it by the iconic diagram from Shannon's classic paper, with the information source as the terroir, and the drinker as the destination.


You can't help but think about Marshall McLuhan's phrase "the medium is the message' in this context!

I'm sure my layers and trapeziums could be used to make this model work better.