7 Box \(A\) contains 8 white balls and 2 yellow balls. Box \(B\) contains 5 white balls and \(x\) yellow balls. A ball is chosen at random from box \(A\) and placed in box \(B\). A ball is then chosen at random from box \(B\). The tree diagram below shows the possibilities for the colours of the balls chosen.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Box \(A\)}
\includegraphics[alt={},max width=\textwidth]{60a9d5d4-0a6a-43e2-9828-03ea2a76ed8a-3_451_874_1774_639}
\end{figure}
- Justify the probability \(\frac { x } { x + 6 }\) on the tree diagram.
- Copy and complete the tree diagram.
- If the ball chosen from box \(A\) is white then the probability that the ball chosen from box \(B\) is also white is \(\frac { 1 } { 3 }\). Show that the value of \(x\) is 12 .
- Given that the ball chosen from box \(B\) is yellow, find the conditional probability that the ball chosen from box \(A\) was yellow.