At work we have weekly training sessions. Each Thursday one of our team members takes one hour to teach a topic to the rest of the team. The themes range from hands-on use of Excel templates over negotiation skills to creating a business case. The respective theme depends on our annual goals, i.e. what we have agreed to learn, and on the interests of the trainers.

About a week ago I gave our team a training on the Monty Hall problem. The problem is easy to understand, yet the correct answer is somewhat unintuitive and explaining it in an understandable way takes some effort. It also shows well how easy it is to get probabilities wrong. Therefore I had the Monty Hall problem as the day’s topic.

I first presented the problem, and to make it more interesting I asked for volunteers to play against me. This way I wanted to get some statistical data to back up my upcoming explanation of the probabilities in the game. Before starting I presented two questions that we would answer during the training:

- Is it profitable to play the game, when the participation costs 0.70 and the potential gain is 1.00?
- Which strategy maximizes the expected winnings? And why?

Here I must mention that I introduced the a version of the game, where there is always a prize behind exactly one door, the prize is never removed and the hosts always opens exactly one empty door at random, but never the player’s chosen door.

Before we started playing, one team member asked me for the goal of the training. I replied that the goals were to:

- Understand the Monty Hall problem and the related probabilities in general.
- To see that probabilities are not always intuitive and thus we should beware when making decision, even if we think we have calculated the risks correctly.

I ended up playing the game with one colleague for ten rounds, so we did not get any statistically meaningful results, but these games already gave us some feeling of the game, how it works and how it feels to be in the situation, having to choose between the doors. After the game I explained the probabilities involved by drawing on a paper the three potential outcomes in a single game and arguing that by changing the initial choice the chances of winning are maximized. This way it was quite easy to convincingly illustrate that sticking to the original door only gives a one in three chance to win.

I think I wan’t able to convince all team members of the importance of understanding the Monty Hall problem or the importance of learning new things outside of your daily business. An in all I was still pleased with how I managed to explain the problem and its essence in a practical and tangible way.

**Afterthoughts**

The following weekend after the training, when I was again on one of my many walls in the woods, four things dawned to me regarding the lessons given by the Monty Hall problem:

- When presenting the Monty Hall problem, it should be thought of as not just as a game of chance, but as a representation of an action containing risk. The game can be thought of as as investment, where the costs and the payoffs determine, whether the investment is profitable.
- Even more importantly, the problem teaches us to concentrate on the essential. The game has two stages: In the first stage the player chooses a door and the host opens an empty door. In the second stage the player may change his initial choice. The game can be represented as a one stage game, where the player only makes a decision between choosing one door or choosing two doors. Due to the rules of the game, the host’s opening one empty door is just trickery used to distract the player. If we omit the door opening and give the player the option of choosing either one door or two doors, the intrinsic probabilities of winning become obvious. Thus, it is necessary to see what is relevant.
- Based on point 2, when teaching or learning, we should try to present the topic or problem from as many aspects as possible. That way it is easier for more people to learn the subject matter. The multiple representations also enable us to recognize a potentially familiar pattern or situation in a completely different context. This way we can apply the different tools and models we learn. E.g. we might, when presented with an offer, see some similarities to the Monty Hall problem and thus know to be careful in making our choice and are also equipped to analyze the offer correctly.
- When teaching, we should emphasize the importance of learning different things and models just to understand how the world might look like. As an analogy, a person with only a hammer is less useful at a construction yard than a person who also has a saw, a grinder, a pen and some paper in his pocket and a crowbar. When we have multiple tools and know how and when to use them, we are better able to act in different situations. With multiple models we are also better equipped to recognize the important things and act accordingly. Sometimes learning something does not provide obvious or immediate benefits, but might later on in life prove to be very valuable.