This is a concept I have always struggled to understand:
When talking about Discrete Probability Functions, we usually have no problem assigning a probability to a single event. For example, when rolling a 6-sided die, we can say that the probability of encountering any of sides will have a certain probability.
However, when talking about Continuous Probability Functions, we are told that that idea of "individual probabilities" does not exist. For example, consider a random variable "X" that has a Normal Distribution with Mean = 0 and Sigma = 1. We can find out the probability of this random variable "X" taking values between 2.9 and 3.1 - but we can not find out the probability of this random variable "X" taking an exact value of 3.
I always wondered - what is the formal reason behind this?
We know that the "Mode" represents the most common number of a sample - and the "Mode" is a single number.
We could then find out the probability of a random variable taking values between "mode + 0.0001" and "mode - 0.0001" (e.g. via an integral) - and try to see if this number converges for smaller and smaller deviations from the "mode", thus finding out the probability of encountering exactly the mode.
I am aware that this logic that I have described is most likely flawed - but could someone please help me understand why the concept of "individual probabilities" does not exist in the continuous sense?