Problems

It has been observed that Professor Ecks proves three types of theorems: 1, those that are correct and new; 2, those that are correct, but already known; 3, those that are false. It has also been observed that, if a certain of her theorems is of type \(i\), then her next theorem is of type \(j\) with probability \(p\low_{ij},\) where \(p\low_{ij}\) is the entry in the \(i\)th row and \(j\)th column of the following array: \[ \begin{pmatrix}0.3 & 0.3 & 0.4\\ 0.2 & 0.4 & 0.4\\ 0.1 & 0.3 & 0.6 \end{pmatrix}\,. \] Let \(a_{i},\) \(i=1,2,3\), be the probability that a given theorem is of type \(i\), and let \(b_{j}\) be the consequent probability that the next theorem is of type \(j\).

1. Explain why \(b_{j}=a\low_{1}p\low_{1j}+a\low_{2}p\low_{2j}+a\low_{3}p\low_{3j}\,.\)
2. Find values of \(a\low_{1},a\low_{2}\) and \(a\low_{3}\) such that \(b_{i}=a_{i}\) for \(i=1,2,3.\)
3. For these values of the \(a_{i}\) find the probabilities \(q\low_{ij}\) that, if a particular theorem is of type \(j\), then the \textit{preceding }theorem was of type \(i\).