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 = 1 − p 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?

Now draw a tree and assign probabilities 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)?

Assuming 0 and 1 represent the two states we get the folloing Markov Chain:

Markov Chain

          |  0  |  1
       ---|-----|----
       0  |  q  |  p
       1  |  p  |  q

if we assume p + q =1 then the transition matrix follows:

\(\left(\frac{q\quad p}{p\quad q}\right) = \left(\frac{1-p\quad p}{p\quad 1-p}\right) = \left(\frac{q\quad 1-q}{1-q\quad q}\right)\)

Probability

The probability would be given by \(p^{2} + q^{2}\)