Sequential selection without replacement

6 Freddie has two bags of marbles.
Bag \(X\) contains 7 red marbles and 3 blue marbles.
Bag \(Y\) contains 4 red marbles and 1 blue marble.
Freddie chooses one of the bags at random. A marble is removed at random from that bag and not replaced. A new red marble is now added to each bag. A second marble is then removed at random from the same bag that the first marble had been removed from.

1. Draw a tree diagram to represent this information, showing the probability on each of the branches.
2. Find the probability that both of the marbles removed from the bag are the same colour.
3. Find the probability that bag \(Y\) is chosen given that the marbles removed are not both the same colour.

3 A box contains 3 red balls and 5 blue balls. One ball is taken at random from the box and not replaced. A yellow ball is then put into the box. A second ball is now taken at random from the box.

1. Complete the tree diagram to show all the outcomes and the probability for each branch. First ball
   Second ball
   \includegraphics[max width=\textwidth, alt={}, center]{7dc85f33-2647-4f73-8093-524b70f99767-04_655_392_688_474}
   \includegraphics[max width=\textwidth, alt={}, center]{7dc85f33-2647-4f73-8093-524b70f99767-04_785_387_703_1110}
2. Find the probability that the two balls taken are the same colour.
3. Find the probability that the first ball taken is red, given that the second ball taken is blue.

5 A bag contains 4 blue discs and 6 red discs. Chloe takes a disc from the bag. If this disc is red, she takes 2 more discs. If not, she takes 1 more disc. Each disc is taken at random and no discs are replaced.

1. Complete the probability tree diagram in your Answer Book, showing all the probabilities.
   \includegraphics[max width=\textwidth, alt={}, center]{48ffcd44-d933-40e0-818a-20d6db607298-4_730_1203_529_511} The total number of blue discs that Chloe takes is denoted by \(X\).
2. Show that \(\mathrm { P } ( X = 1 ) = \frac { 3 } { 5 }\). The complete probability distribution of \(X\) is given below.

   \(x\)	0	1	2
   \(\mathrm { P } ( X = x )\)	\(\frac { 1 } { 6 }\)	\(\frac { 3 } { 5 }\)	\(\frac { 7 } { 30 }\)

3. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

4 A bag contains 5 red discs and 1 black disc. Tina takes two discs from the bag at random without replacement.

1. The diagram shows part of a tree diagram to illustrate this situation. \section*{First disc}
   \includegraphics[max width=\textwidth, alt={}]{e23cb28b-49e5-436a-942d-e6320029c634-3_264_494_479_550}
   Complete the tree diagram in your Answer Book showing all the probabilities. \section*{Second disc}
2. Find the probability that exactly one of the two discs is red. All the discs are replaced in the bag. Tony now takes three discs from the bag at random without replacement.
3. Given that the first disc Tony takes is red, find the probability that the third disc Tony takes is also red.