Graeme Archer: Untesting times

Graeme Archer, Hackney activist and ConservativeHome regular, has his own blog.

Look, I’m not about to go off on one about “dumbing down” in general, or launch a tirade about the pitfalls of popular culture. I’m not an opera buff and I’ve never been the ballet. Though we did go and see Don Giovanni a few months ago. (If you ask me, three hours is a long time to sit and wait for the bit where the Devil appears and yells “Don GioVANNi” towards the end, quite the best bit.) But there’s something very wrong happening in the world of light entertainment, something which I think is getting close to abuse.

Maybe it’s the statistician in me (what a coy way of putting it, why is it so hard just to come out with? My name is Graeme Archer and I AM a STATISTICIAN. There). There’s a quiz show on in the early evenings on Channel Five called “The Grid” which appals me; and “Deal or No Deal” isn’t just an irritating vehicle to herald the Return of the Edmunds, it’s symptomatic of something rotten, a festering nexus at the heart of our culture, where poverty of education finds its manifestation through trick-playing on the public.

The set up of The Grid is a high-tech studio where “players” stand in front of a large cube, each face of which contains (the same) grid of I think 25 squares i.e. 5 rows and 5 columns. The three-dimensional aspect of this set up is quite irrelevant to what occurs on the programme, part and parcel of the con-trick which then takes place, a con-trick which can only happen because the participants are either unaware of what they’re doing, or are willing to suspend their belief in anything rational for the sake of appearing on television.

The programme has multiple rounds and a few variants but it boils down to the following: behind each of the 25 squares is either a sum of money or a black square to indicate that the player loses all or a portion of the sum thus far accumulated. In the final, for example, there might be 4 such “bombs” lurking. The participant names a square by its grid reference, which is then revealed to contain either money or a bomb. Before the sequence begins, the participant is told how many squares contain the hidden bomb. Thus with 4 bombs, the participant has a 4/25 chance of picking a bomb on his or her first go, then 4/24 on the second go, and so on …

What’s my problem with this? It is that, instead of the participant saying “I want A1 (the top left square), then A2 if I am not bombed out, then A3…” and so on, the player, and the “quiz”master, and the audience, all commit a joint deceit of pretending (or believing) that there is some sort of strategy which can lead to a better outcome than simply reading out the codes of the squares in any random sequence. You have to see it to believe it. It’s appalling. Somehow, the people involved have induced a drama from reading out numbers at random, and are imbuing this process with pseudo-meaning. In fact, if you think about it, human participation in this process is irrelevant. One could simply tell the producers that one would pick 5 squares at random, without leaving the comfort of one’s home.

“Deal or no deal” is perhaps slightly less egregious, in that I’m willing to accept that you might require some degree of mathematical sophistication to understand that at any stage in its tortuous process, you can quickly work out the expectation of gain for any box (identical) and whether or not it’s worth continuing, or stopping to take the “banker”’s offer. But it’s been on our screens for months now, and still people seem to think that there’s something almost magical happening. No-one in the media has written an article to expose the essentially void centre of these programmes, which can only succeed – and they do succeed – in a population that has become both frighteningly innumerate and is willing to imbue random processes with a quasi-spiritual meaning.

Does it matter? Yes it does and it should worry Conservatives. Firstly, it’s a very clear signal that something is going wrong with basics in education. Secondly, it is a clear sign that popular culture is becoming non-aspirational: people no longer win prizes for demonstrating proficiency in anything, but merely by having the “skill” to select numbers at random. Thirdly, if people don’t understand that the expected value of one choice from two boxes is half of their total, then explaining the acts of piracy enacted on their pension funds by Brown is not going to be as straightforward as we might anticipate.

For these reasons, George Osborne was right to call for the introduction of financial education in schools. I might want more, and ask for an investigation of what is being taught in the maths syllabus these days. When I was a wee laddie in Scotland, we called these “skills” arithmetic, and nobody left school without a full command of it.

Bah humbug? I know I know I’m a sad old statistician and I should maybe just let people enjoy a harmless bit of fun. But ITV at night-time is pumping out programmes like these, all designed to remove money from gullible viewers. Someone makes money from this – via premium phonelines - or it wouldn’t happen. And so I think there’s something morally wrong happening here, and I hope that our MPs will vote for action to end it – I know that a Commons committee has begun a relevant investigation, but they were quietly sniggered at by the (I assume innumerate) commons sketch writers in the quality press. These latter should have been ashamed of themselves.

On a lighter note! Not all statistical reasoning is intuitive. For a bit of fun, suppose you’re on a quiz show and are faced with three doors, A, B and C. Behind two of the doors is a copy of The Guardian, behind the third is a lovely new iPod. You pick a door and get whatever’s behind it. You want to win the iPod, and being a good statistician, pick a door at random, “A”, say (without clutching your head and shouting “Come ON” at the audience). The host (who knows what lies behind each door) doesn’t open door A, but opens either door B or door C to reveal a copy of The Guardian, and says to you “Do you wish to stay with door A, or switch?”. What ought you to do to maximise your chance of winning? I’ll post the answer later.

Comments

[Winchester whisperer]

Switch of course!

[Scotty]

Graeme, a very astute article. Was nodding in agreement about the quality of game shows and remembering such wonders as "Blockbuster's". You then highlighted the very reason that I believe education has been undermined and used as a tool in Labour's ever increasing war against statistics for political expedience.
"and ask for an investigation of what is being taught in the maths syllabus these days. When I was a wee laddie in Scotland, we called these “skills” arithmetic, and nobody left school without a full command of it."
Back in my long ago school days I remember that great emphasis was placed on maths as subject that was essential if you planned to go to university whatever degree you took, but arithmetic was seen as an O grade that all children should have when leaving school whatever career path they took. It was in essence a required life skill like reading and writing.
"all commit a joint deceit of pretending (or believing) that there is some sort of strategy which can lead to a better outcome than simply reading out the codes of the squares in any random sequence."
But then when we are at the stage when parents require a glossy booklet to understand their children's school report card then something is wrong! My parents commented on the fact that they could see at a glance just what we were achieving in school and why did anyone need to turn it into a product similar to a DIY instruction manual.

[Automated Robot]

Switch.

But this has always been counter-intuitive to me and I can never remember why. Perhaps I just cannot understand the reasoning.

Your choice is independent of what lies behind the remaining doors (your name isn't Schroedinger is it?) and indeed the door that was opened, unsolicited, by the host.

[Annabel Herriott]

I would not dream of watching that crap in the first place.

[Denis Cooper]

Confronted with three doors, behind one of which there is an iPod, and behind two of which there are copies of the Guardian, I pick a door. The host then opens one of the other two doors and reveals a copy of the Guardian. If he had revealed the iPod, presumably by the rules I would have left the stage with just the copy of the Guardian which was behind the door I had chosen. But as he revealed a copy of the Guardian behind the door he opened that door now becomes irrelevant and in fact it, rather than me, could be removed from the stage.

So in effect I'm now playing a new game, in which I'm confronted with two doors, behind one of which there is an iPod, and behind the other of which there is a copy of the Guardian. If I merely picked a door at random I would I have a one in two chance of choosing the iPod. However by dint of my psychic powers I had already ascertained that the door I chose previously is the one concealing the iPod, and it came as no suprise to me when the host opened one of other doors and revealed a copy of the Guardian, because I already knew that to be true of both of the doors I had rejected.

Obviously therefore I stay with my original choice.

[Sally Roberts]

Good grief, Graeme! My mind has been totally boggled after reading this - but then I have always been totally innumerate!! You can educate me on numeracy skills - and I'll have to educate you on opera.... It isn't the Devil who calls out to Don Giovanni in the opera - it's the ghost of The Commendatore (Donna Anna's father) who the Don killed in the first scene!!!

[Peregrine]

I think the reasoning behind a switch goes something like this:
If the host only opens a second door if you have chosen incorrectly (and assuming that you don’t have this information) then by switching you are guaranteeing getting the iPod (assuming that you aren’t looking for that plum Local Authority Diversity Co-ordinator job in the Guardian).
If the host always opens a second door and it is always a second prize, then you have an evens chance of getting the prize you want which ever door you choose and switching will not affect this chance.

[Winchester whisperer]

It is crucial to understand the game show host is not opening a door at random. If your first choice was wrong, then the game show host has no choice in which door he opens.The reason this is crucial is because, if the game show host were truly opening a door at random, then your switching doors WOULD be simply a 50/50 chance.As for demonstrating that switching doors is not 50/50, I know of no better way than this: 2/3 of the time you will pick the wrong door first. Then the game show host will have 2 doors left: one of them right, one of them wrong. He will open the wrong door, leaving the third door, which (if you picked the wrong door first) is the right door WITH 100% CERTAINTY.

So 2/3 of the time the third door will be right with 100% certainty.

[Denis Cooper]

"2/3 of the time you will pick the wrong door first. Then the game show host will have 2 doors left: one of them right, one of them wrong. He will open the wrong door, leaving the third door, which (if you picked the wrong door first) is the right door WITH 100% CERTAINTY."

That's true, but 1/3 of the time you will have picked the right door. Suppose that's door A, as Graeme suggested. The host knows that, and he knows that he can open either door B or door C, and in either case you would be proved wrong if you switched. As the host has those two choices, that's two distinct routes by which you would lose the iPod if you then switched from your original choice - by switching to the unopened B, or to the unopened C.

Starting with all three boxes, the alternatives are (G = Guardian, I = iPod):

1. A = I, B = G, C = G
2. A = G, B = I, C = G
3. A = G, B = G, C = I

After the host has opened either B or C, if necessary knowingly picking whichever of the two conceals a Guardian rather than the iPod, the four equally probable alternatives for the two remaining boxes are:

1a. A = I, B = G
1b. A = I, C = G
2. A = G, B = I
3. A = G, C = I

In effect you are now playing a new game with two doors, one of which conceals an iPod and one of which conceals a Guardian, and the fact that the host has shown you that the third door concealed a Guardian is irrelevant because that box is now irrelevant.

[John Allen]

The Mathematicians and Scientists who understood Graeme Archer's article will understand why the Speed Camera scheme is a waste of time. In fact, may be increasing accidents.
The innumerate will believe any spin as they have no basis for undertaking the simplest check.
"If it looks right then it probably is right. If it does not then you can bet your boots it is not." Ones capability of looking depends on ones knowledge - science/maths/stats are essential.

[sjm]

While I sympathise with the gist of this argument, I'm not happy with the idea of Parliament intervening.

What really worries me is that so many idiots like those who participate in and watch this garbage actually have the vote!

I don't think you should get the vote until you've passed an IQ and General Knowledge test. That should reduce the average voting population from around 63k to 1k (on a really good day).

[Denis Cooper]

Looked at from another point of view, when the host chooses a door and opens it he tells you nothing useful that you don't already know.

You knew from the start that with two Guardians and one iPod, whichever door you picked there would certainly be a Guardian lurking behind AT LEAST one of the other two doors, and you also knew that the host knew what was behind each of the doors and would avoid opening a door if the iPod was behind it.

The host has ceremoniously opened one of those two doors, and lo and behold there's a Guardian behind it. But that merely confirms what you already knew, that there would be a Guardian behind AT LEAST one of those doors, and that the host would know which door to avoid if there was a Guardian behind ONLY one of those doors.

By opening that door he's given you no additional information about what lies behind either the door he chose NOT to open, or the door which you picked, which is what matters to you if you're now wondering whether to switch. You're really none the wiser for his bit of showmanship.

[Graeme Archer]

I'm very very sorry for writing such a dull piece today! Thanks for these comments guys - I'm guessing that not many other people get as stressed by random quiz shows as I do :-0)

Sally you're quite right, I know not nearly enough about opera. It's a good new year's resolution. This is what happens when you waste money taking someone like me to the opera, with an attention span of a fruit fly :-0). It's still the best bit of the night though isn't it?

Solution to the Doors problem

Winchester Whisperer is quite right of course - somewhat counter-intuitively, one should always switch in the situation outlined above. There's a very good explanation of why on Wikipedia, which saves me explaining. All inductive reasoning should always be Bayesian you know.

Use this URL to see the explanation:
http://en.wikipedia.org/wiki/Monty_Hall_problem

[David Sergeant]

I have been meandering on the fringe of politics a very long time and I remember the good old days when Labour were socialists and workers voted for them because they were workers. In those days every time Labour were losing an argument a Labour leader would come out with utter rubbish to keep morale up. Most workers bought the rubbish but, since everyone else knew it was rubbish, it was left in middair. About then I learnd that TV advertising was aimed at a memtal age of eleven - this, perhaps explaining why Labour did as well as it did.

Graeme Archer's article struck a cord with me about to-day and Newlab. Say, in parliament time after time we are told Blair or Brown came out on top but when you carefully follow what was said the Tory spkesperson has used careful argument (which rarely gets reported) and Blair has used playground rubbish (clunking fist) which gets reported as a winning saying. It seems that the lowering standards of education have not only affected more people, even the media now is only interested in punch and judy - even younger than eleven!

[james miles]

It does matter that the host opens a door. It does seem counter-intuitive.
To get the answer you need to construct your decision tree for the scenario where you have complete choice (no door is opened).Initially you have 3 choices (1 win, 2 lose), and then from each one, another 2 choices (2 win, 4 lose). If you were merely to choose another door at random 4 out of 6 choices would be bad ones.
By opening one door, the host 'forces' the statistics in his favour. Now if you swap doors, you have a 2 out of 3 chance of winning, but on the face of it, you only seem to have a 50:50 chance, so you might not bother...and in time the host will win more often than the sucker member of public.

[Denis Cooper]

Well, I would have been a sucker member of the public, and I must admit that I'm still having a problem getting my head around the explanation!

[Simon Walterd]

Of course as a group that works for a political party you should all be grateful that the majority of the UK population does watch and enjoy these game shows. Otherwise modern politics could never work as it does since we would all see through the deceit and ineptitude. As Billy Connolly said - no one who wants to become a politician should ever be allowed anywhere near politics. Go on - flame me:-)

[matt wright]

Well they may be pretty silly programmes but there have been pretty silly programmes and games that have captured people for many years. Its not really a new phenonemon,

Matt

[David Gillies]

There seems to be some sort of evolutionary glitch in our resoning skills when it comes to assessments of probability and the like. The 'Monty Hall' problem you mention here is a good example. Another is the Wason test. One variant goes like this: you have four cards. Each has a number on one side and a letter on the other. The cards are laid on the table thus: G | M | 8 | 5. Your task is to ascertain whether the rule "if a card has a G on one side, it has an 8 on the other side" is true. Which cards do you need to turn over? More than 50% of people get this worng. But if the values on the cards are recast in a form that onvolves the detection of cheating, the success rate goes way up. Let's recast the problem: cars with odd number plates must park on the left side of the road. The cards now read whether cars have odd numbered plates and which side of the road they are parked. They are: odd number | even number | left side | right side. Which ones do you turn over to verify compliance with the rule? Obviously the first and the fourth. This is the same answer as the abstract case with letters and numbers, but Leda Cosmides and John Tooby have hypothesised that our greater facility with the second example is an evolutionary relic of our ability to enforce social contracts.

[Alan]

Heaven forfend that anyone should act on superstition and irrational impulses; because that never happens in the Conservative party, right? Religion must just be a figment of my over active imagination!

I think these games shows are a bit of fun and should be taken as such. There was a link to a Telegraph article on this site, not long ago, about the proliferation of popular tv programmes like the X factor which do promote the importance of merit so I think you're considering this trend somewhat out of context Graeme. I would have stuck with door A by the way. I suppose I should be disenfranchised now!

[Peter]

As I read this thread I just got one point, what was the point with it? We have had so many news articles on elections(South Ossetia, Bahrain and UAE to name a few) and the like and Graeme feels as a statistician he should have ago at deal or no deal. As another statistician get a grip will you and tells us what you feel about any of the above elections(South Ossetia is a good one to start with) please rather than have ago at a program which is actually good at showing the mindset of people at a given time?

[Ali Cat]

Graeme's article is as usual full of information and very thought provoking. However I liked the comment from "Posted by: Peter Dec 19th". He condems Graeme's comments as I think 'being too frivolous' and says he himself is a statistician. Pity he didn't do a bit more english when at school maybe then he'd know the difference between 'a go' and 'ago'
Ali cat

[Peter]

Hopefully as Archer is within your email account Ali Cat, you not actually Graeme in disguise. What information also was passed on by the original comment by Graeme do you find interesting and also did nothing about elections in the forementioned countries even merit a cimment from you, what a shame?

[Graeme Archer]

Just for the record, "Ali Cat" is not me!