8) A certain calculating machine uses only the digits 0 and 1. It is supposed to

Question

8) A certain calculating machine uses only the digits 0 and 1. It is supposed to transmit one of these digits through several stages. However, at every stage, there is a probability p that the digit that enters this stage will be changed when it leaves and a probability q = 1p that it won’t. Form a Markov chain to represent the process of transmission by taking as states the digits 0 and 1. What is the matrix of transition probabilities?

9)For the Markov chain in Exercise 8, draw a tree and assign a tree measure assuming that the process begins in state 0 and moves through two stages of transmission. What is the probability that the machine, after two stages, produces the digit 0 (i.e., the correct digit)? What is the probability that the machine never changed the digit from 0? Now let p = .1. Using the program MatrixPowers, compute the 100th power of the transition matrix. Interpret the entries of this matrix. Repeat this with p = .2. Why do the 100th powers appear to be the same?

I need the answer for number 9 it depends on number 8, please help me with that. Thank you

Explanation / Answer

8)

This problem is identical to the “telephone” question from class. Our states in this case are the digits 0 and 1. Our transition matrix is P = (q p p q )

9)

All powers of P = (1 0 0 1 ) are just (1 0 0 1 ) . For P = (0 1 1 0 ) , even powers are $0 1 1 0 % and odd powers are (1 0 0 1 )

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