/sci/ - Science & Math

A vampire can attack two people every night.
Every time a vampire attacks a human, there is a chance [math]C[/math] they will kill the human.

A human attacked by a vampire but who was NOT killed will instead turn into a Servant of the vampire.
A Servant can attack one human every night.
Every time a Servant attacks a human, the Servant will attempt to turn the human into a fellow Servant. This process has a [math]0.9[/math] chance of succeeding, i.e. there a [math]0.1[/math] chance of the process failing and the human dying. Humans who die after being attacked by a Servant stay dead.

A human who was killed by the vampire has a [math]0.01[/math] chance of coming back to life as a ghoul. Given a sample of 100 such eligible corpses killed by a vampire, it would take [math]t_{1\backslash2}=30[/math] days for 50 of them to rise up as a ghouls. A ghoul will then feast on the flesh of the dead to grow in power. Every night, there is a [math]0.5[/math] chance of a ghoul encountering another ghoul. When two ghouls meet, one eats the other and gains the other's total power. It takes the combined power of 666 ghouls for a ghoul to turn into a proper vampire.

Once a ghoul turns into a vampire, he will seek out his "parent vampire" and defeat him. Let us define [math]S_{total}[/math] to be the number of Servants the original vampire has at the time of his defeat.

Express [math]S_{total}[/math] as a function of [math]C[/math].