question 1

1. Imagine a world where souls exist in only three lifeforms: ores [who have a 5% chance of dying each year}, humans [who have a 1% chance of dying each year] and elves {who have a H.197: chance of dying each year}. When a ore, human or elf dies, their soul transfers to either a orc, human or elf and they are reincarnated according to the following rules: I Q''} of orcs, when they die, are reincarnated as ores. The rest. are reincarnated as humans. I Bad humans (30% of humans] when they die are reincarnated as orcs. Reasonably good humans {ll 971'} of humans] when they die are reincarnated as humans. Exceptionally good humans {l' of humans} when they die are reincarnated as elves. I Most elves {99%} eventually turn evil and when they die are are reincarnated as orcs hut asmall percentage (1%} stay good and when they die are reincarnated as elves. 0n Using a timestep of one year, and a state vector x\" = hn where on, is the percentage [of the n total population of orcs, humans and elves] of orcs at timestep n, h\" is the percentage of humans at timestep n and on is the percentage of elves at timestep 7:, do the following: [a] [B marlrs] Describe a Markov chain for this system if at timestep [I there are equal numbers of orcs, humans and elves. [h] [3 marks] What percentage of souls exist in humans after l years? {c} [3 marks] [11 the very long term [after untold aeons] what percentage of souls exists in each of the three lifeforms