There Just Might Be A Gremlin In Your House.

Discrete Mathematics Level 5

40 year old Billy Peltzer gets a Mogwai from his father for old times' sake. This adds to the collection of Mogwai he's received over the years, bringing him to a total of 20 Mogwai. One night, when Billy is about to go to bed to get rest for his business trip, his young son Willy kindly asks to play with the Mogwai. Too tired to care, Billy permits the request on the condition that Willy not feed the Mogwai after midnight, that he not get them wet, and that he not shine any bright light upon them. Willy marches out the door to the Mogwai house (similar to a doghouse) in the backyard and has some fun with the Mogwai. Willy, sleepy like his father, passes out in the Mogwai shed without sealing the latch on the cupboard, leaving the snacks unsecured. Overnight, one of the Mogwai binges on cookies and muffins and turns into a gremlin. Willy wakes in a cold sweat to find the converted Mogwai (and his father has already left for the business trip).

Gremlins and Mogwai interact such that once every minute a Gremlin can choose another individual and turn it into a Gremlin. Similarly, a Mogwai can choose an individual and turn it into a Mogwai. If a Mogwai chooses another Mogwai, or a Gremlin chooses another Gremlin, nothing changes. The relative chance that a given Gremlin is chosen to convert an individual (versus an individual Mogwai being chosen) is \(r_{G} = \frac{19}{17}\).

Question: What is the probability that all 20 are Gremlin by the time Willy's dad returns two weeks later?

Details and assumptions

  • For our purposes, two weeks is an infinite amount of time .
  • Obviously, once all 20 become either Gremlin or Mogwai, there can be no further changes and the population is permanently Gremlin or Mogwai.
  • Only one Mogwai or Gremlin is chosen to convert an individual per minute.
  • If there are \(n\) Gremlins at a given time, the chance that a Gremlin will be chosen to convert a Mogwai is \(\displaystyle\frac{n r_{G}}{nr_{G} + \left(20-n\right)}\frac{20-n}{20}\).
  • If there are \(m\) Mogwai at a given time, the chance that a Mogwai will be chosen to convert a Gremlin is \(\displaystyle\frac{m}{m + r_{G}\left(20-m\right)}\frac{20-m}{20}\).

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