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Splitting bugs over the generations
By Chirag Bhatt on March 2, 1998:

My problem concerns the splitting of bugs. According to the question there are three kinds of bugs-happy, sad and blank. In the next generation the following changes take place:

  • Every happy bug splits into a sad on and a blank one.
  • Every sad one splits into two happy bugs.
  • Every blank bug splits into a happy bug and a sad bug.
We are supposed to investigate the bugs in different generations. I have managed most of the questions but one has baffled me. Itsays ``Investigate and work out the probabilities of choosing each type of bug when one is taken from any generation''. I have worked to 14 generations but am not sure how to work this question out.

Please help.

Chirag Bhatt
By Jeff on March 2, 1998:

A way to get somewhere with this question is to note that the number of bugs always doubles each generation. This means that it might be a good idea to consider the proportions of the different types of bugs at each generation rather than the total number.

We could write these as a "vector", for example, (0.1,0.2,0.7) would mean that 10% of the bugs are happy, 20% blank and 70% sad. Call this vector at generation n, X(n). Thus, if you start with just one happy bug, then X(0)=(1,0,0) following the rules you gave, what would X(1), X(2) etc. be?

Can you write down a general rule to find X(n+1) if you know X(n)?

Suppose that the proportions after a long time (i.e. large number of generations) settle down to staying roughly constant. When they are constant, what very simple equation do we get between X(n) and X(n+1) for very large n?

A further investigation might be to determine which starting positions will lead to a settling down of the proportions after a long time.