Edit Problem

Year

Paper

Question Number

Course

Section

Worksheet Citation (for copying)

Click the copy button or select the text to copy this citation for use in worksheets.

Problem Text

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 $j$. 
\begin{questionparts}
\item Explain why $b_{j}=a\low_{1}p\low_{1j}+a\low_{2}p\low_{2j}+a\low_{3}p\low_{3j}\,.$
\item Find values of $a\low_{1},a\low_{2}$ and $a\low_{3}$ such that $b_{i}=a_{i}$
for $i=1,2,3.$
\item 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 $i$. 
\end{questionparts}

Solution (Optional)

Preview

Problem

Solution

Cancel

Current Ratings

Difficulty Rating: 1700.0

Difficulty Comparisons: 0

Banger Rating: 1554.3

Banger Comparisons: 6