# There Just Might Be A Gremlin In Your House.

**Discrete Mathematics**Level 5

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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