Page last updated at 15:39 GMT, Monday, 18 May 2009 16:39 UK

The odds of Deal or No Deal

By Richard Knight

Is Channel 4's gameshow Deal or No Deal a useful maths lesson, or a haven for superstition?

Deal or No Deal
The sum is fixed at the beginning but the Banker's offer changes

Contestants on Deal or No Deal face 22 numbered boxes - each containing prize money, ranging from just one penny to a quarter of a million pounds.

The player holds just one of those boxes, not knowing how much money is inside. They choose a series of boxes to be opened and discarded.

If all the other boxes are opened the contestant will receive the amount that is in their own box, be it 1p or £250,000.

But during the process of opening the boxes, a "Banker" interrupts by telephone to offer a sum in order to get the contestant to stop the game.

The contestant must judge whether it is likely, bearing in mind the boxes that have been opened and discarded, that the amount in their box is higher than the amount being offered by the banker.

More or Less is on BBC Radio 4 on Fridays at 1330 BST
Listen to the programme here

Opening a box with a very small amount inside - coloured blue - is good, because that low amount is eliminated. Opening a box with a high amount inside - coloured red - is bad, for the same reason.

The distribution of prizes is highly skewed. When the game starts, the mean average of the prizes received by contestants is almost £26,000.

But only five out of 22 boxes contain more than that, so the majority of players will have less than £26,000 in their own box.

The mathematician player

And half the boxes contain £750 or less. Players tend to agonise over which boxes to open. But the truth is they might as well choose at random.

The interesting decision is the one that gives the show its title - whether to say "deal" or "no deal" when the Banker intervenes.

The anonymous Banker communicates only via a Bakelite telephone - he offers the contestant cash in exchange for the unknown sum in his or her box.

Noel Edmonds and More or Less presenter Tim Harford
Mathematics does not rule everyone's choices

The mathematician-player basing their decision on probability would only exchange the unknown quantity inside their box for the Banker's offer if that offer were higher than the value they could expect to win by chance.

The value they could expect to win by chance could be calculated by dividing the total amount of money still in the game by the number of remaining boxes.

The Banker's offer will tend to be lower than the "expected value" of the prize, but it is a sure thing. What is more, the Banker's offer will frequently be higher than the amounts in most of the remaining boxes.

The Banker's offers start low and edges closer to fair value as the game goes on, tempting contestants to stay in.

But of course, your typical player is probably not a mathematician.

Economists have studied the behaviour of the contestants. One paper was entitled Deal or No Deal? Decision Making under Risk in a Large-Payoff Game Show.

One idea is that people become less cautious about risk after some early bad luck.

And many do not get bogged down in probability calculations.

One contestant, who was not very successful, suggested she relied on luck.

"My game was based around my grandfather Patrick who was born in Ireland. And St Patrick's Day was 17th March. So I had 16 and 17 left.

"Now when you've got a lucky box, you can either think it's going to be lucky - so this is when I need to take this box out. But actually it would have been lucky if I'd kept it because the fifty grand was in there. It can be lucky for both reasons."

Those involved with the show acknowledge that contestants use a mixture of calculation, superstition, and brinkmanship.

"It's about all those things," says presenter Noel Edmonds. "It's got a massive psychology behind it."

If you watch Deal or No Deal with a mathematical eye, you'll pretty soon see that many of the contestants have strange beliefs and little interest in figuring the odds. But perhaps that's not the fault of the game - we can be funny creatures when it comes to numbers.

More or Less is produced in association with the Open University.

Below is a selection of your comments.

I like the show but still feel it is down to pure good luck or bad luck. What it does demonstrate is greed as when a contestant states a £20,000 prize would be life-changing for them. During the box opening they receive an above £20,000 offer and because they have red boxes higher they turn down the offer which is real money in their hands. True to sods law they usually end up with a paltry sum and say it doesn't matter but obviously are gutted. I would love to be on the show but then again playing at home as to being in the studio would probably alter any tactics I say I would use.
Tim Mcmahon, Wales

Isn't relying on expected value a bit over-sensitive? My reflex is to use median price, and jump to the bankers offer if it is above median price. Why? Because I could be wrong but I thought that is a more reliable or risk averse measure, because it is less influenced by the big boxes suddenly dropping out.
Josh W, Swansea, Uk

The bankers offer should not be compared to the expected value as suggested here since this would imply comparing the current offer to the expected payoff given that the player continues playing to the end and receiving the value in the players box. The format of the game is far more complex since there will be offers made at set periods between the current offer and the end of the game. The fair value of the game should be computed using a multinomial option valuation model. This is possible since there are only finitely many different states of the game.
John & Rajan, Warwick Business School

There's no such thing as a lucky box just as there is no such thing as a lucky coin, but something has to guide the players to make their decisions - and often makes for better television. A concern I have with this show is when players state that they 'believe' a box will contain a high amount because it's their child's birthday, or their partner has told them this number beforehand, or that they had a dream...when the box turns out to house little more than a few quid, do the contestents blame fate or the aforementioned child, partner, dream? One hopes not. On the other hand, perhaps they deserve it for believing in such superstitious nonsense. Luck is a commodity of hindsight, luck has no memory.
Dean, Exeter

Whilst at university, I used to call this "Pick a Box" - many of my peers (usually sociology students) watched it religiously, and claimed it was a game of immense skill - they refused to accept my explanation involving probability. Fools.
PB, London

I watch the program in complete disbelief when people say "I have a system!" who then defy logic and start spouting they mechanics of their system. I can see the excitement, see the anticipation and adrenaline involved and how that affects decisions, however you might as well work from one end to the other, weighing up the real question "If I continue, am I more or less likely to out-perform the offer". I guess I'd never be accepted for the show because I would apply nothing but cold logic in the face of money.
James B, Sheffield, UK

What isn't considered here is the impact of previous games on the current one. The position of the different value boxes changes due to variable factors in each new game. The boxes are delivered randomly to each position. Consider in five positions the £250,000 appeared in position 1 in the previous game. If I was picking the boxes to open I would pick position 1 because in my mind the chance of the 250k box appearing in the 2-5 positions is higher. People will argue that each game there is a 1 in 22 chance of a box being in anyone position. However previous games should be considered. It is chance but you can study the probability and variables to aide in your decisions of luck.
Jon, Wiltshire, UK

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