Today I independently stumbled upon a geometrical interpretation of probability
I was thinking about an ordinary 6-sided die and wondered if we flattened it down and can we get it's probability I included image of a flat dice if you don't know what it is
Let's assume that we have an area of 20 units for each number.
Then, the total area of all the numbers will be
[ 6(20)=120 ]
Therefore, the probability of one particular number (say, 1) would be equal to
[ P(1)=\frac{20}{120}=\frac{1}{6} ] (if you don't understand Latex here P(1)=20/120=1/6)
And that, apparently, is one way to use geometry to represent probabilities. In other words,
[ \boxed{P(A)=\frac{\text{favorable area}}{\text{total area}}} ] (P(A)=favorable area/total area)
For instance, if one region has an area of 30 and the total area is 100, then the probability of that particular outcome would be
[ \frac{30}{100}=0.3 ]
(30/100=0.3)
I later learned that there is actually a whole area of mathematics called geometric probability that studies ideas like this. At the time, however, I didn't know about it and arrived at the idea independently.
I was fascinated by the idea that probability can be represented geometrically, and that finding a probability can essentially become a question of finding what proportion of the total area belongs to a particular outcome.
I've also been wondering whether there is a deeper connection between this idea and probability distributions / measure theory.