Problem 5 (Markov Process for Word Types). Assume that each word that a particular poet writes can be classified as a noun, verb, adjective, adverb, preposition, or conjunction, and that the poet has written at least word in the past.

In the future, whenever a poet intends to write a word, they will look at the last word they wrote, and roll a 20-sided die that is equally likely to land on any of the whole numbers from 1 to 20, inclusive.

If the last word they wrote was a noun, and they roll any number from 1 to 15, inclusive, then they write a verb. If they roll a number from 16 to 20, inclusive, then they write a conjunction.

If the last word they wrote was a verb, and they roll a 1 or 2, then they write a noun. If they roll a number from 3 to 6, they write an adjective. If they roll from 7 to 12, they write an adverb. If they roll from 13 to 16, they write a preposition. If they roll from 17 to 20, they write a conjunction.

If the last word they wrote was an adjective, and they roll from 1 to 14, they write a noun. If they roll from 15 to 20, they write another adjective.

If the last word they wrote was an adverb, and they roll a 1, then they write another adverb. If they roll a 2 to 10, then they write a preposition. If they roll an 11 to 20, they write a conjunction.

If the last word they wrote was a preposition, and they roll a 1 to 5, then they write down a noun. If they roll a 6 to 13, they write an adjective. If they roll a 14 to 20, then they write a conjunction.

If the last word they wrote was a conjunction, and they roll a 1 to 5, then they write a noun. If they roll a 6 to 10, then they write a verb. If they roll an 11 to 17, then they write down an adjective. If they roll an 18 to 20, then they write down a preposition.

QUESTION B:

1. Assume that the most recent word that the poet wrote down was an adjective. What is the probability that the word after the next word that the poet writes

   will be a preposition?

QUESTION C:

1. Assume that the most recent word that the poet wrote down was a noun. Suppose that, henceforth, we keep track of the proportion of words written by the poet that are nouns, verbs, adjectives, adverbs, prepositions, and conjunctions. In the long run, what proportion of words written will be of each type?