# Curse of Dimensionality

In machine learning, like many fields of computer science, often involves high dimensional points, and high dimension spaces have some surprising probabilistic properties.

A random *value* $X_i$ is a Uni(0, 1).

A random *point* of dimension $d$ is a list of $d$ random values: $[X_1 \dots X_d]$.

A random *value* $X_i$ is close to an edge if $X_i$ is less than 0.01 **or** $X_i$ is greater than 0.99. What is the probability that a random value is close to an edge?

A random *point* $[X_1, X_2, X_3]$ of dimension $3$ is close to an edge if *any* of it's values are close to an edge. What is the probability that a $3$ dimensional point is close to an edge?

A random *point* $[X_1, \dots X_{100}]$ of dimension $100$ is close to an edge if *any* of it's values are close to an edge. What is the probability that a 100 dimensional point is close to an edge?