Deal Or No Deal: investigating gameshow maths

My first memory is of my dad getting into a fight with Noel Edmonds on the set of Telly Addicts. He had gotten so bored with the endless trivia that he fell asleep during one question, and woke up during another, and ran onto the stage to complain that David Jason wasn’t The Fugitive. If he found a colourful 80s gameshow boring, good lord knows what he makes of random guesstheboxathon Deal Or No Deal. He would probably implode into a coma.

But is it really just a game of chance? Unfortunately, try as you might, you can never get away from that agonising itch that it’s all just a completely random number guessing game. Albeit one with a delusional man who has phone conversations with a non-existent financial warlord, and 4000 gallons of tears. Delve deep into the mathematics though and instead of finding some formula, it turns out it’s even more down to probabilities and statistics than you thought.

Let’s start with the basics. You have a 1/22 chance of picking the biggest prize at the start (although equally you have a 1/22 chance of picking any value).  If you consider success to be one of the five biggest, it’s a 5/22 chance. OK, let’s try to be clever. What’s the chance that you won’t pick the highest box on each turn? Assuming the first box you pick (to keep) isn’t the £250,000 prize, the chance of continually avoiding the star prize decreases as the boxes are removed. Taking the product gives us…

Damn. There’s no way of getting around it, the probability of “winning” Deal or no Deal is 5/22, about 23%. This is, incidentally, the percentage of Noel Edmonds’ face that is beard.

There are however some twists to try and obfuscate the chance. The banker, and the last box.

Every game begins the same way. Someone is chosen at “random” from the box keepers to play. The amount in their box is what they win, unless they swap at the end. My first thought, as I’m sure was yours, was to think of former Miss Earth’s Highest IQ, Marilyn vos Savant.

The Monty Hall Problem she postulated the answer for is a counter-intuitive (that’s maths speak for “no one really understands it unless they have a calculator and a beard”) probability trick loosely based on the game show Let’s Make A Deal. You’re probably lying if you say you remember it, because it was never shown here. In this thought experiment, you are faced with three doors, behind one of which is a car. The other two, goats. You pick a door but don’t open it, and then the host opens one of the other doors, to reveal a shiny new goat. The problem is this: do you stick, or twist? If you said “it doesn’t matter you idiot, there’s only two doors left so it’s 1/2”, then surprisingly you’d be wrong.

Since the probability of you picking a winner first time is only 1/3, the probability either of the other doors is a winner is 2/3. But even though the host has revealed a non-winning door, it doesn’t change that your door has only a 1/3 probability of having the car; the host opening door simply guarantees that the door he didn’t open has the 2/3 chance all to itself. Therefore you should switch doors. Of course, the real answer is you should take the goat, because as Goat Simulator has shown us, goats can provide tremendous entertainment.

On the face of it, then, you should switch boxes at the end of Deal Or No Deal, because you’ve acquired new information throughout the game by eliminating the boxes. Unfortunately, this doesn’t apply – Edmonds himself doesn’t ever reveal anything about the boxes you haven’t opened. The closest he ever comes to giving any information is via the banker, and all he ever does is make an offer based on the value remaining. Without extra information on the boxes you haven’t opened, the odds of getting the swap right are 50/50, just like you’d expect. So back to square one then. 

After the first box is chosen, our contestant picks boxes at random to throw away, with each one comes an outpouring of grief and tears normally reserved for an X Factor sob story. A number is flashed up on screen, wonderfully colour coded so we know which are the small ones and which are the big ones without having to think too hard. Then another random box is chosen, and more tears. This continues until there are no more boxes left or someone cries themselves into dehydration.

Of course, this was the way the game was designed. In fact, the Gambling Commission met with the producers a few years ago because they suspected that there was in fact no skill involved and was therefore a lottery. Lotteries are banned in this country with one obvious exception, so Noel came close to being thrown back into television obscurity. This rule, incidentally, is why daytime TV shows run stupid quizzes with questions like “Paris is the capital of which country?  A: France, B: Tom Cruise or C: Kwik Save”, because apparently this involves some sort of skill. Much in the same way as breathing, or not falling over. Of course, the show is still on the air – the producers didn’t even attempt to prove it’s not a lottery, they just pointed out it can’t be gambling if the contestants don’t pay to enter.

So is there any way to get one over on them? Perhaps. Noel Edmonds and his imaginary phone calls.

The banker turns up at the end of each round of box opening via telephone to hurl insults and then, crucially, to offer the contestant a deal, based on the current values left on the board. This is where it becomes a game of skill, albeit a fairly simple one if you’re familiar with game theory. The banker makes an offer based on the root mean square (RMS) value v of the remaining boxes, which involves squaring the values, taking the mean, and then square rooting again.

Basically it’s a fancy term for a middle weighted average, so that any of the extreme values don’t skew the average too much. The banker tends to offer an approximation of this, based on how far along the game is and whether the player is “playing well” i.e. if they’re providing some decent entertainment value like crying profusely or wearing a loud shirt. If it’s early in the game or you’re doing things like whipping a calculator out and explaining what a root mean square is, he’ll probably offer you a couple of paper clips and a slap in the thorax. Thus, the strategy is to provide enough entertainment value by pretending to think people have any control of any of the boxes, while thanking everyone on the planet and crying. This still relies on chance of course – you need to open a few blues boxes to weight the average higher or else any deal will be rubbish.

Either way, the ultimate decision on whether to deal is whether or not you’re “beating the mean”. If you’re offered substantially lower than the RMS of the remaining boxes, tell the banker to bank off. If he offers more, you should probably take it, especially if you’ve only got a few reds left. If you’ve only got one red left, don’t get greedy, just take the offer, even if you think it’s too low. The best you can hope is that you’ve been enough of a loud crying sadsack that he takes pity on you and gives you a good offer. Fortunately for the rest of us, the set of crying sadsack doesn’t overlap well with mathematicians, so “entertainment” ensues.

So what counts as a success?  Well, the expected value of the boxes before the game is given by the sum of the values, weighted by the chance of finding them.

“But that’s just the average of all the prizes!” I hear you yell, and of course you’re right. This works out to about £25,712.12. But since this is still a game of chance, you have to weigh getting this offer versus the chance of getting a higher box. There are only five boxes with higher values than this, and since each box has the same chance, your chance of winning is 5/22, or 23%. Which is what I already told you.

This article first appeared in May 2014.