In this question you must show detailed reasoning.

A disease that affects trees shows no visible evidence for the first few years after the tree is infected.

A test has been developed to determine whether a particular tree has the disease. A positive result to the test suggests that the tree has the disease. However, the test is not 100% reliable, and a researcher uses the following model.

• If the tree has the disease, the probability of a positive result is 0.95.

• If the tree does not have the disease, the probability of a positive result is 0.1.

\begin{enumerate}[label=(\alph*)]
\item It is known that in a certain county, $A$, 35% of the trees have the disease. A tree in county $A$ is chosen at random and is tested.

Given that the result is positive, determine the probability that this tree has the disease. [3]
\end{enumerate}

A forestry company wants to determine what proportion of trees in another county, $B$, have the disease. They choose a large random sample of trees in county $B$.

Each tree in the sample is tested and it is found that the result is positive for 43% of these trees.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item By carrying out a calculation, determine an estimate of the proportion of trees in county $B$ that have the disease. [4]
\end{enumerate}

\hfill \mbox{\textit{OCR H240/02 2023 Q14 [7]}}