Transfer between containers

1. Two bags, \(\boldsymbol { X }\) and \(\boldsymbol { Y }\), each contain green marbles (G) and blue marbles (B) only.

• Bag \(\boldsymbol { X }\) contains 5 green marbles and 4 blue marbles
• Bag \(\boldsymbol { Y }\) contains 6 green marbles and 5 blue marbles

A marble is selected at random from bag \(\boldsymbol { X }\) and placed in bag \(\boldsymbol { Y }\)
A second marble is selected at random from bag \(\boldsymbol { X }\) and placed in bag \(\boldsymbol { Y }\)
A third marble is then selected, this time from bag \(\boldsymbol { Y }\)

1. Use this information to complete the tree diagram shown on page 7
2. Find the probability that the 2 marbles selected from bag \(\boldsymbol { X }\) are of different colours.
3. Find the probability that all 3 marbles selected are the same colour. Given that all three marbles selected are the same colour,
4. find the probability that they are all green. 2nd Marble (from bag \(\boldsymbol { X }\) ) \section*{3rd Marble (from bag Y)} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{1st Marble (from bag \(\boldsymbol { X }\) )} \includegraphics[alt={},max width=\textwidth]{c316fa29-dedc-4890-bd82-31eb0bb819f9-07_1694_1312_484_310}
   \end{figure}

1. Two bags, \(\mathbf { A }\) and \(\mathbf { B }\), each contain balls which are either red or yellow or green.

Bag A contains 4 red, 3 yellow and \(n\) green balls.
Bag \(\mathbf { B }\) contains 5 red, 3 yellow and 1 green ball.
A ball is selected at random from bag \(\mathbf { A }\) and placed into bag \(\mathbf { B }\).
A ball is then selected at random from bag \(\mathbf { B }\) and placed into bag \(\mathbf { A }\).
The probability that bag \(\mathbf { A }\) now contains an equal number of red, yellow and green balls is \(p\). Given that \(p > 0\), find the possible values of \(n\) and \(p\).

4. Three bags A, B and \(\mathbf { C }\) each contain coloured balls. Bag A contains 4 red balls and 2 yellow balls only.
Bag B contains 4 red balls and 1 yellow ball only.
Bag \(\mathbf { C }\) contains 6 red balls only. In a game
Mike takes a ball at random from bag \(\mathbf { A }\), records the colour and places it in bag \(\mathbf { C }\). He then takes a ball at random from bag \(\mathbf { B }\), records the colour and places it in bag \(\mathbf { C }\). Finally, Mike takes a ball at random from bag \(\mathbf { C }\) and records the colour.

1. Complete the tree diagram on the page opposite, to illustrate the game by adding the remaining branches and all probabilities.
2. Show that the probability that Mike records a yellow ball exactly twice is \(\frac { 1 } { 10 }\) Given that Mike records exactly 2 yellow balls,
3. find the probability that the ball drawn from bag \(\mathbf { A }\) is red. Mike plays this game a large number of times, each time starting with the bags containing balls as described above. The random variable \(X\) represents the number of yellow balls recorded in a single game.
4. Find the probability distribution of \(X\)
5. Find \(\mathrm { E } ( X )\) Bag B
   Bag C \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Bag A} \includegraphics[alt={},max width=\textwidth]{29ac0c0b-f963-40a1-beba-7146bbb2d021-13_739_1580_411_182}
   \end{figure}