Puzzle #1

[Magicman]

Both A and C have an equal 50/50 chance of having the car behind them. The solution therefore is to toss a lucky coin.

I rewfer you to

http://betterexplained.com/articles/...-hall-problem/

Magicman

[Empirical]

The obvious, 'intuitive' answer is that it makes no difference whether you stick with door A or change to door C. It's also wrong; the explanation isn't so easy to understand or explain, but here goes.

At the start, there is an equal chance that the Rolls is behind A, B or C:

thus, the chances are A = 1/3, B = 1/3 and C = 1/3

that is, A + B + C = 1 which we can rewrite as

A + (B+C) = 1, so

(B+C) =2/3

we now discover that B is not = 1/3, but is 0

therefore, (0 +C) = 2/3

therefore, C = 2/3

[and A remains as 1/3]

therefore, by swapping, you increase your chance of the Rolls from 1/3 to 2/3.

This may not be the 'purest' mathematical approach, but the reasoning is correct. Curiously, this result was first given by a female mathematician; curiously, perhaps, because women aren't supposed to think logically; there are very few female mathematicians. There was a major outcry when she published this, many refused to accept it; one would only accept it when shown a computer simulation.

There was such a simulation recently on a BBC programme, though I forget its name. Wikipedia has a good article on the Monty Hall problem; related problems such as conditional probability are really hard going. "The prosecutor's fallacy" is a sort of example of this sort of problem, again there's an article on Wikipedia; the ultimate disaster was the case of Sally Clarke, wrongly convicted of murder through mathematical, probability error.

https://en.wikipedia.org/wiki/Prosecutor%27s_fallacy

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

It's not at all surprising, therefore, that politicians and journalists are easily foxed by even the simplest of mathematical problems and probabilities; you only have to look at the garbage they spout about sex work to realise just how little they really understand.

[Dr. Ivory]

I have not read the links - I will do so but have not yet. Conditional probability is difficult. However, in the precise example given of the doors, I still don't think it applies. You argue:
" A + (B+C) = 1, so

(B+C) =2/3

we now discover that B is not = 1/3, but is 0

therefore, (0 +C) = 2/3

therefore, C = 2/3"

What I don't agree with in the above is that if B=0 then it is no longer the case that

A+(B+C) = 1

rather, it is now the case that

A+C = 1

To show you what gets me about this, tell me what is wrong with the following counterexample

A, B and C have an equal chance of being correct.
I pick A

A+B+C=1

(A+B) = 2/3

Now B=0

A = 2/3

So I should stick with A.

All you have to do is put brackets in wherever you like.

That said, I should read the links :-) I vaguely remember this monty hall thing from Philosophy of Mathematics class.

[Dr. Ivory]

Ah jaysus, just read the Wikipedia page. Heh, heh, the oul brain is getting really slow. But I get it now: The choices made at T1 (between a/b/c) and at T2 (between a/c) are connected and not discrete events because the behaviour of the chat show host is rule based and not random. Furthermore the host operates under a key combination of two rules - 'don't open the door the contestant chose' and 'don't open the door with the car behind it.' In two out of three possible scenarios for the t1 choice, the combination of those two rules, means that the door neither chosen nor opened has the car behind it. You're right, I was wrong.

That was fun.

[Empirical]

Ah jaysus, just read the Wikipedia page. Heh, heh, the oul brain is getting really slow. But I get it now: The choices made at T1 (between a/b/c) and at T2 (between a/c) are connected and not discrete events because the behaviour of the chat show host is rule based and not random. Furthermore the host operates under a key combination of two rules - 'don't open the door the contestant chose' and 'don't open the door with the car behind it.' In two out of three possible scenarios for the t1 choice, the combination of those two rules, means that the door neither chosen nor opened has the car behind it. You're right, I was wrong.

That was fun.

You're quite right, and perhaps I should have explained this in more detail.

Glad you enjoyed it!