The probability is the area / 確率は面積である Associate Professor Makiko Sasada, Mathematics


“The probability is the area”. Associate Professor Makiko Sasada,
Mathematics ”How was the mathematics studied at the university?”

The mathematics studied at the university are very different from those up to high school. The opportunity to calculate was gone.
I came to get a lot of new concepts. I learned to abstract concepts that “what does it mean to have holes?”, “what does it mean to measure quantity?”, “what does symmetry mean?” I also obtained the theorem by making it general and universal. And I considered how to prove it. And I considered how to prove it. At the beginning I felt it was very difficult, but on the other hand I felt it was very interesting. Research in mathematics is to learn how to view or how to cut things and abstract it. I thought that having a rich perspective was the study of mathematics. “Specifically, what is an interesting part of mathematics?”

The most impressive class is probability theory class. Until high school I was learning problems like “the probability that 1 comes out in the dice” at the class. But at the university class, I was very surprised to hear “the probability is the area”. Actually, “the probability is the area”. For example, let’s consider a method of knowing the area of a circle without using the circular constant Pi. Draw a square that fits snugly around the circle. And I hit a lot of points in it and check the ratio with the number entered into the circle. If it is a circle with a diameter of 1, the area is 1 because its outer square is 1 x 1. By counting the number of points which entered the circle, you can see the ratio of the area of the square to the area of the circle. That is, you can find the area of the circle. The data you see is an experiment that hits a lot of points with Excel obtained for the Pi. It turns out that it converges to 3.14 gradually while repeating points. Similarly, you can know the area even in the heart shape. Draw a square around the heart and hit a lot of points. The area ratio of the square and the heart can be known by the number entered into the heart of the whole, so the area of the heart can be obtained. In other words, the area and the probability are inseparable from each other. Let’s think about weather forecast. We call the correlation table of sunny, cloudy and rainy areas “the probability”, but we interpret it as “the area” in modern mathematics. I think that what we can define like this is a great discovery. “What can be obtained from “the probability” and “the area”?”

Once, “the probability” was a difficult concept that bothers mathematicians. The definition of “the probability” was very ambiguous. When talking about the probability of this year’s stock price becoming over some price, people have different ideas. That is, it depends on the information and subjectivity the person has. Mathematicians were struggling to formulate subjective and ambiguous “probabilities”. By deciding “the probability has specific rules as well as the area”, it had become possible to express probability completely with objective and universal words. With this, subjective and ambiguous ideas could be eliminated. This was defined around the mid-1910s about 100 years ago. So that, in a long history of mathematics it is a very new way of thinking. I think it can be said that the 20th century is the era of probability.
With this, the mathematics related to probability developed explosively and greatly. As a result, it is applied to various fields such as physics, biology and economy.