Two-stage sampling with replacement

Questions where a ball/disc is selected from one container, placed in another container, then a second selection is made from the second container (Q6384, Q6696).

3 questions

CAIE S1 2024 June Q3
4 marks
3 Box \(A\) contains 6 green balls and 3 yellow balls.
Box \(B\) contains 4 green 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 \(B\).
  1. Draw a tree diagram to represent this information, showing the probability on each of the branches.
    [0pt] [4]
    \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-06_2727_38_132_2010}
    The probability that both the balls chosen are the same colour is \(\frac { 8 } { 15 }\).
  2. Find the value of \(x\).
CAIE S1 2020 March Q6
6 Box \(A\) contains 7 red balls and 1 blue ball. Box \(B\) contains 9 red balls and 5 blue 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.
  1. Complete the tree diagram to show the probabilities. Box \(A\)
    \includegraphics[max width=\textwidth, alt={}, center]{f7c0e35d-1889-4e5b-b094-f467052a66cf-08_624_428_667_621} \section*{Box \(B\)} Red Blue Red Blue
  2. Find the probability that the two balls chosen are not the same colour.
  3. Find the probability that the ball chosen from box \(A\) is blue given that the ball chosen from box \(B\) is blue.
CAIE S1 2013 June Q7
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}
  1. Justify the probability \(\frac { x } { x + 6 }\) on the tree diagram.
  2. Copy and complete the tree diagram.
  3. 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 .
  4. Given that the ball chosen from box \(B\) is yellow, find the conditional probability that the ball chosen from box \(A\) was yellow.