Temporal state in probability — “Monty Hall problem” and more
A solution to avoid confusion
I had not yet gone to university and was in my senior year of high school when I first heard about the “Monty Hall problem” from a friend. Considering my preparation in probability and mathematics lessons, solving this problem was easy for me and I wondered to my friend why he found this problem difficult to solve. But later I realized that when Mrs. “Marilyn vos Savant” published the correct answer to this problem for the first time, even people educated in mathematics did not accept her answer and accused her of ignorance in mathematics and probability.
Decades have passed since the design of the “Monty Hall problem” and every time other types of this type of problem are proposed in the field of probability, such as the “Three Prisoners problem”, which is another form of the previous solution. In these years, I have been asked about this problem many times and I have personally explained this problem and its solution to dozens and maybe hundreds of people. In many cases, people who were curiously listening to the correct solution did not understand my solution or doubted the solution. I tried again to change the way of expression and even the type of argument, but I still felt that the audience could not accept the correct answer. Many of these people were extremely cleverand had studied in the best universities and had very high academic degrees in the field of engineering, medicine, etc. So what is the issue? Why do so many people doubt the solution?
Recently, I was listening to a reputable psychiatry podcast, and the psychiatrist used the “Monty Hall problem” many times to explain an important point and I realized that a lot of scientific work has been done on this topic. Of course, the topic of this text is not psychology and I have little knowledge in this field. But due to the importance of the topic, I decided to share my experience in expressing complex probabilities. That is, in the following, I want to introduce a simple method (Temporal state in probability) that my audience has understood the subject better with this method.
Before I explain about the “Temporal state in probability”, please watch the video below. The link below is an old video from the popular “Numberphile” channel. In this video, “Monty Hall problem” and its correct solution are explained using the image. In the following, I assume that the reader has seen this video.
Temporal State
Usually, the information about the probability is said in a few sentences. The first sentence is the same as the first “State”. So everything must be recorded carefully so that the first “State” is completely defined. For example, for the “Monty Hall problem”, it is determined that the probability of each door is one third.
The second sentence is the changes you need to make to the first “State” to create the second “State”. In addition, you should pay attention to whether the second sentence is related to all the members of the collection or only some of the members. For example, in “Monty Hall problem”, only the information added in the second sentence is related to the two doors not selected and has nothing to do with the selected door. So the probability of the selected door does not change and remains one third. But on the other side, i.e. the probability of unselected doors, a big change takes place and it is determined that only one door is left and the probability of this door is equal to the total probability of unselected doors (i.e. two-thirds). Because anyway, the sum of all possibilities is always equal to the number one.
But for example, if you are familiar with “Three Prisoners problem”, the information that prisoner “A” gets by begging from her jailer has nothing to do with herself, so there is no change in her chances of being pardoned.
The third sentence (if any) is the changes you need to make in the second “State” to create the third “State”. It is obvious that the fourth sentence and the following include the same definition.
The above method of expressing and explaining the problem in probability can help a lot and I hope it will be useful for you too. By the way, if you look carefully in the video above, somewhere in the video she assumes that the total number of doors is one hundred and explains this situation as well. This method (that is, changing the number of members) is also a very good addition to the explanation.
