some math equations ;)

[ctb]

$P$(observe k events) =

Here the parameter $\lambda = \frac{M}{S}$ is the expected number of hashes for $M$ k-mers at a scaled of $S$.

$P$(k hashes for M k-mers with scaled S) = ${\exp(-M / S) \over k! } ({M \over S })^k $

$P$(empty sketch for M k-mers => $k=0$) = $e^{-M / S}$

[ctb]

For any regions $R_A$ and $R_B$ with identical FracMinHash sketches at scaled $S$, the probability that the differential containment is smaller than a fraction $f$ of $R_A$ is:

$P({| R_A - R_B | \over |R_A|} < f) = \exp(- {f | R_A | \over S })$