## Archive of posts with tag “probability”

### This is what will happen after you get a haircut

Denote the length distribution of one’s hair to be $f(l,t)$, where $l$ is hair length, and $t$ is time. Considering that each hair may be lost naturally from time to time (there is a probability of $\lambda\,\mathrm dt$ for each hair to be lost within time range from $t$ to $t+\mathrm dt$) and then restart growing from zero length, how will the length distribution of hair evolve with time? It turns out that we may model it with a first-order PDE.

- Categories: math
- Tags: calculus, probability, pde

### The longest all-$1$ substring of a random bit string

Given your probability of breaking the combo at each note, what is the probability distribution of your max combo in the rhythm game chart? I considered the problem seriously!

- Categories: math
- Tags: long paper, rhythm game, algorithm, probability