Doing a PhD is not only about doing research, contact with others about your research is very important. Often this means going to a conference or talking to other mathematicians at the university, but sometimes it means explaining your research to children. This was the idea of the children’s square at the vierdaagsefeesten in Nijmegen (parties around the walking event where children and adults walk long distances during 4 days): reach as many children as possible and show how fun science can be.

Winning with mathematics Since my own research on entropy in noncommutative geometry is a bit too advanced for such a medium, and we had a group of mathematicians working together, we focused on mathematics in general. Under the topic of how to win using mathematics (‘‘Winnen met Wiskunde’’ in Dutch) we’re able to show children how mathematics can help us in daily life, for example when playing a game show.

Monty Hall problem One of the things we set out to do was to play and explain the Monty Hall problem, which was a part of the game show Let’s Make a Deal, which was on television from the sixties. So the situation is as follows: you, the contestant, want to win a car and do this by correctly identifying behind which of three doors this car is. You lose if you are left with one of the other doors at the end of game, behind these doors are goats. If you only had to make a choice between 3 doors with no extra information, the chance of correctly identifying the door would be 1/3, so if this were the game it would be easy to see that no tactics are involved. However, in this game the host will give you some extra information. After you picked out one of the doors, the game host will open one of the doors with a goat, and then asks you whether you want to switch to the one other door remaining. When we did this with children we saw that most don’t want to switch from their original door to the remaining door. When asking them why they made their choice, you often get answers like: I don’t know, I just have a good feeling about this. The first thought that most adults have is that now both remaining doors have a 50 percent chance of hiding the car, so you might as well remain with your original choice.

Tracking choices So what we did was keep track of what happened: switched vs stayed and won vs lost. We saw that most people ended up in the stayed and lost category, and that the amount of people in this category was roughly twice as much as those staying and winning the game. So why doesn’t this represent the 50-50 chance that people expect?

This is because this 50-50 chance doesn’t use all the information given, this would only be the case if there were 2 doors from the beginning. That the host will never open the door with the car behind it is information in disguise. While the following might sound counterintuitive, let me explain why switching gives you a 66 percent chance of winning:

Assume that the car is behind door 1, we have the following possibilities:

• You pick door number 1, without loss of generality door 2 is opened by the host. If you stay you win, but changing to door 3 would mean you lose.
• You pick door number 2, the host must open door number 3. You win by changing to door number 1 and lose by staying at door 2.
• You pick door number 3, the host must open door number 2. You win by changing to door and lose by staying.

So we see that if you stay, you only win in 1 out of 3 cases, and changing gives the car in 2 out of 3 cases. But it turns out that this explanation is not only hard to understand for children, also adults are confused about this results. When the first mathematicians came up with this explanation, there was great controversy about its validity. But doing the game many times, we find that this explanation indeed gives the right result.

Take-away What makes this such a good example is that we can actually do the game itself, but also get a feeling for how counterintuitive mathematics can be. I also like it because you start from something we can easily see happening, and then we have to find an explanation for it and have to argue why this would be the case. While it is quite complicated to grasp immediately, you see that many are convinced to try this theory and after multiple times of losing, will win on their first try with this new method. The ability of children to accept information and immediately try it out, is why doing outreach for children is so rewarding.