It’s 3pm, I’ve done all my work for the day, I’ve (probably not) done my daily exercise, it’s nice weather outside and it would be a lovely evening to go to the pub with my mates. But, oh wait, we’re in lockdown. So, what do I do? Put on the telly of course. It’s prime gameshow time; first Tenable, then Tipping Point, then The Chase, then Pointless, then Richard Osman’s House of Games… If I’m lucky there might even be some Mastermind or University Challenge on after dinner!

For many of us, at the moment, this may be a rather common (and pretty sad) highlight of daily life. It may seem like you’re passing the hours by filling your head with random, niche and highly useless knowledge, but under the surface, gameshows may have a lot more to offer than you might initially think.

Gameshows are inherently based around decision-making. Whether it’s an individual answering general knowledge questions, or a team choosing which teammate should answer a question, or perhaps deciding whether or not to play for a higher amount of money – a player’s success in a gameshow is determined by their ability to make the ‘right’ decisions. But what is the ‘right’ decision? Is there an optimal strategy for success? Could you ever outsmart the gameshow producers?

In this article, I will explore these questions and those closely related to them. I will first provide an overview of the research that has been done on this topic, with some classic examples of mathematics in gameshows. Then, we’ll look at some modern-day examples of these kinds of problems and how you can approach them. Finally, we’ll discuss the limitations of using mathematics within this context and key takeaways.

**Where it all started: The Monty Hall Problem**

Try this: Google “mathematics in gameshows”. Sure enough, articles on the Monty Hall Problem will come up as some of the top results. So, what’s all the fuss about? Why did this particular show attract so much attention from mathematicians? Let’s take a look.

Back in the 70’s there was an American gameshow called “Let’s Make a Deal” hosted by a rather famous man called Monty Hall. The concept was simple enough; there were three doors, numbered 1, 2 and 3, respectively. Behind one was your dream car and behind the other two were “zonks” – things you didn’t really want.

So, the contestant would pick a door, say door number 1 for simplicity. Monty would then walk over to the two doors that the contestant didn’t choose (doors 2 and 3) and open one of them. Without fail, behind the door he opened was a “zonk”. After this, Monty turned around to the contestant and asked, “Would you like to stick with your decision of door 1 or switch to the other remaining unopened door?”.

The contestants’ strategy varied; some chose to stick, some chose to switch. They believed that either way their chance of winning the car was 50:50 – so it didn’t matter what they did, their chance of winning would be the same. Actually, this wasn’t the case at all. In fact, to give the greatest chance of winning the car – the contestant should ALWAYS switch. But why?

When the contestant makes their initial decision of which of the three doors to pick, there is quite clearly a 1/3 chance that the car is behind their chosen door. This means that there is a 2/3 chance that it is behind either of the other two doors that the contestant did not pick. But then Monty shows you that behind one of these doors is “zonk”. So, actually, there is now a 2/3 chance that the car is behind this final door that the contestant did not choose and Monty did not open. That leaves us with a 1/3 chance of winning by sticking with the originally chosen door and a 2/3 chance of winning by switching to the other unopened door. By switching, the contestant DOUBLED their chance of winning the car.

This problem is actually a very basic application of the centuries-old Bayes’ Theorem of conditional probability, which describes the way that probability of events changes when we are given prior information regarding them and what happens if we update this information. It is vital to remember that, in the above problem, we make a number of assumptions (and if these are not met, we cannot draw the same conclusion for the optimal strategy). These are as follows:

- Monty will always open a door.
- Monty never opens the door you have chosen.
- Monty never opens the door with the car behind it.
- The car is equally likely to be behind any door.
- Given a choice of doors, Monty chooses at random.

Simple yet counterintuitive, this problem sparked the interest of mathematicians (and, unsurprisingly, gameshow producers) world-wide. In fact, many modern-day shows exhibit similarities to the Monty Hall Problem. Let’s take a look at an example…

**Could you beat The Banker: Deal or No Deal?**

In the UK version of this classic game show, a contestant chooses one of the twenty-two briefcases. Each of the briefcases contains a cash sum ranging from 1 penny to £250,000. By a process of elimination, the contestant aims to win as much as possible, either by gambling on having a high amount in their own briefcase or by making ‘The Banker’ – the games’ hidden operator who also has no prior knowledge of what is in each briefcase – offer a considerable cash sum for what is inside the box. If the contestant decides to play until the end (where just their own and one other box remains), they are asked by the show’s host if they would like to swap their box for this other remaining box.

Again, we have the same idea of revealing what’s behind doors/boxes, only with more steps and the addition of deciding whether to accept The Banker’s offer – right? So, by the same logic as the Monty Hall problem – if you play until the end, you should always swap your original box for the other box. Actually, this is incorrect. At no point is anything actually revealed about the boxes you haven’t opened, so the chances of making a ‘successful swap’ for a higher amount of money are just 50:50 as you would expect. So, the game is just purely down to chance? Again, wrong.

The introduction of The Banker’s offer at various steps in the game is what makes Deal or No Deal interesting. The maths behind it is actually relatively simple to any mathematician familiar with game theory. At the various stages of the game, all The Banker does is make an offer based on the mean root squared (MRS) value of the remaining unopened boxes.

To calculate the mean root squared value, you really do just do as it says on the tin:

- Square root the values of the unopened boxes
- Take the sum of these values
- Divide by the number of unopened boxes
- Square this number.

$$\text{The Banker’s offer} \approx \left( \frac{\sum_i^n \sqrt{v_i}}{n} \right)^2$$

where $n$ is the number of unopened boxes, $v_i$ is the price of the unopened box $i$, and the sum ranges over $i = 1,2,\ldots,n$.

By using this variation of the mean average statistic, The Banker can basically make sure that the extreme values (1p or £250,000, etc.) don’t skew the average too much. Other factors that influence The Banker’s offer include how far along the game is and whether the contestant is entertaining the audience – a little less mathematical.

So, could you beat The Banker? Well, it depends on how generous The Banker’s offer is and how good you are at calculating a mean root squared (MRS) average in your head. Essentially, if The Banker makes an offer above the MRS of the remaining unopened boxes, you should make a deal and take his offer. If The Banker makes an offer well below the MRS of the remaining unopened boxes, well then you tell him to get lost.

Essentially, you can maximize your chances of winning a relatively high amount of money by making sensible decisions about whether to take The Banker’s offer or not, but other than that, the game really is down to luck.

**A guide not a guarantee**

We’ve established that, while mathematics could help you to decide whether it would be sensible to make a ‘deal’ with The Banker or whether you should switch doors in the Monty Hall problem, it will never GUARANTEE success.

Say you switch doors, as advised by the mathematics, in Hall’s “Let’s Make a Deal” – there’s still 1/3 chance that the car is in fact behind the door you originally chose. Switching, while more likely to get you that dream car, won’t ALWAYS get you that dream car.

Similarly, in “Deal or No Deal” – The Banker may make you an offer higher than the root mean squared of the remaining unopened boxes, in which case, the statistically sensible decision would be to make a deal. But that’s not to say that what’s inside your box is less than the amount that you made a deal to trade your box for. I guess the point of this article is: don’t think you can go on a gameshow and swindle the producers out of a large sum of money using your abilities as a mathematician alone. After all, TV and game shows are designed to entertain the audience, so maybe going on Deal or No Deal with your notes on root mean square averages and a calculator isn’t the best idea!

Emilia is a UEA Mathematics graduate, Norwich local, and incoming Consulting Development Analyst at Accenture. Maths-wise, she is particularly interested in statistics and computing, as well as their respective applications to the finance industry. Outside of spending hours of her life puzzling over numbers, she also enjoys a good ol’ pub quiz, shopping, and traveling with her mates.