Sampling without replacement from bags/boxes

7 Box \(A\) contains 6 red balls and 4 blue balls. Box \(B\) contains \(x\) red balls and 9 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 \(B\).

1. Complete the tree diagram below, giving the remaining four probabilities in terms of \(x\).
   \includegraphics[max width=\textwidth, alt={}, center]{217c5a58-2966-4b86-b3b6-9d1676d2979c-12_688_759_484_731}
2. Show that the probability that both balls chosen are blue is \(\frac { 4 } { x + 10 }\).
   It is given that the probability that both balls chosen are blue is \(\frac { 1 } { 6 }\).
3. Find the probability, correct to 3 significant figures, that the ball chosen from box \(A\) is red given that the ball chosen from box \(B\) is red.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 Sam and Tom are playing a game which involves a bag containing 5 white discs and 3 red discs. They take turns to remove one disc from the bag at random. Discs that are removed are not replaced into the bag. The game ends as soon as one player has removed two red discs from the bag. That player wins the game. Sam removes the first disc.

1. Find the probability that Tom removes a red disc on his first turn.
2. Find the probability that Tom wins the game on his second turn.
3. Find the probability that Sam removes a red disc on his first turn given that Tom wins the game on his second turn.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4．A bag contains 64 coloured beads．There are \(r\) red beads，\(y\) yellow beads and 1 green bead and \(r + y + 1 = 64\) Two beads are selected at random，one at a time without replacement．
（a）Find the probability that the green bead is one of the beads selected． The probability that both of the beads are red is \(\frac { 5 } { 84 }\)
（b）Show that \(r\) satisfies the equation \(r ^ { 2 } - r - 240 = 0\)
（c）Hence show that the only possible value of \(r\) is 16
（d）Given that at least one of the beads is red，find the probability that they are both red．

6. At the start of a gameshow there are 10 contestants of which 6 are female. In each round of the game, one contestant is eliminated. All of the contestants have the same chance of progressing to the next round each time.

1. Show that the probability that the first two contestants to be eliminated are both male is \(\frac { 2 } { 15 }\).
2. Find the probability that more females than males are eliminated in the first three rounds of the game.
3. Given that the first contestant to be eliminated is male, find the probability that the next two contestants to be eliminated are both female.
   (3 marks)