Multi-stage with stopping condition

4 To gain a place at a science college, students first have to pass a written test and then a practical test.
Each student is allowed a maximum of two attempts at the written test. A student is only allowed a second attempt if they fail the first attempt. No student is allowed more than one attempt at the practical test. If a student fails both attempts at the written test, then they cannot attempt the practical test. The probability that a student will pass the written test at the first attempt is 0.8 . If a student fails the first attempt at the written test, the probability that they will pass at the second attempt is 0.6 . The probability that a student will pass the practical test is always 0.3 .

1. Draw a tree diagram to represent this information, showing the probabilities on the branches.
2. Find the probability that a randomly chosen student will succeed in gaining a place at the college.
   [0pt] [2]
3. Find the probability that a randomly chosen student passes the written test at the first attempt given that the student succeeds in gaining a place at the college.

7 In a television quiz show Peter answers questions one after another, stopping as soon as a question is answered wrongly.

• The probability that Peter gives the correct answer himself to any question is 0.7 .
• The probability that Peter gives a wrong answer himself to any question is 0.1 .
• The probability that Peter decides to ask for help for any question is 0.2 .

On the first occasion that Peter decides to ask for help he asks the audience. The probability that the audience gives the correct answer to any question is 0.95 . This information is shown in the tree diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{e7e0fcbe-ab96-4292-b3ad-c57b74f15301-3_394_649_1779_386}
\includegraphics[max width=\textwidth, alt={}, center]{e7e0fcbe-ab96-4292-b3ad-c57b74f15301-3_270_743_2010_1023}

1. Show that the probability that the first question is answered correctly is 0.89 . On the second occasion that Peter decides to ask for help he phones a friend. The probability that his friend gives the correct answer to any question is 0.65 .
2. Find the probability that the first two questions are both answered correctly.
3. Given that the first two questions were both answered correctly, find the probability that Peter asked the audience.

7 A bag contains three 1 p coins and seven 2 p coins. Coins are removed at random one at a time, without replacement, until the total value of the coins removed is at least 3p. Then no more coins are removed.

1. Copy and complete the probability tree diagram. First coin
   \includegraphics[max width=\textwidth, alt={}, center]{43f7e091-9ae7-4373-a209-e2ebdba5260f-4_350_317_1279_568} Find the probability that
2. exactly two coins are removed,
3. the total value of the coins removed is 4p.

8 A game at a charity event uses a bag containing 19 white counters and 1 red counter. To play the game once a player takes counters at random from the bag, one at a time, without replacement. If the red counter is taken, the player wins a prize and the game ends. If not, the game ends when 3 white counters have been taken. Niko plays the game once.

1. (a) Copy and complete the tree diagram showing the probabilities for Niko. \section*{First counter} \includegraphics[max width=\textwidth, alt={}, center]{c985b9cc-a202-4d5d-a6b3-591b0560f570-4_293_426_1231_532}
   (b) Find the probability that Niko will win a prize.
2. The number of counters that Niko takes is denoted by \(X\).
   (a) Find \(\mathrm { P } ( X = 3 )\).
   (b) Find \(\mathrm { E } ( X )\).

3 Candidates applying for jobs in a large company take an aptitude test, as a result of which they are either accepted, rejected or retested, with probabilities \(0.2,0.5\) and 0.3 respectively. When a candidate is retested for the first time, the three possible outcomes and their probabilities remain the same as for the original test. When a candidate is retested for the second time there are just two possible outcomes, accepted or rejected, with probabilities 0.4 and 0.6 respectively.

1. Draw a probability tree diagram to illustrate the outcomes.
2. Find the probability that a randomly selected candidate is accepted.
3. Find the probability that a randomly selected candidate is retested at least once, given that this candidate is accepted.

4. A training agency awards a certificate to each student who passes a test while completing a course.
Students failing the test will attempt the test again up to 3 more times, and, if they pass the test, will be awarded a certificate.
The probability of passing the test at the first attempt is 0.7 , but the probability of passing reduces by 0.2 at each attempt.

1. Complete the tree diagram below to show this information.
   \includegraphics[max width=\textwidth, alt={}, center]{70137e9a-0a6b-48b5-8dd4-c436cb063351-08_545_1244_639_340} A student who completed the course is selected at random.
2. Find the probability that the student was awarded a certificate.
3. Given that the student was awarded a certificate, find the probability that the student passed on the first or second attempt. The training agency decides to alter the test taken by the students while completing the course, but will not allow more than 2 attempts. The agency requires the probability of passing the test at the first attempt to be \(p\), and the probability of passing the test at the second attempt to be ( \(p - 0.2\) ). The percentage of students who complete the course and are awarded a certificate is to be \(95 \%\)
4. Show that \(p\) satisfies the equation $$p ^ { 2 } - 2.2 p + 1.15 = 0$$
5. Hence find the value of \(p\), giving your answer to 3 decimal places.
   \includegraphics[max width=\textwidth, alt={}, center]{70137e9a-0a6b-48b5-8dd4-c436cb063351-09_2261_47_313_37}

1. Three bags, \(A , B\) and \(C\), each contain 1 red marble and some green marbles.

Bag \(A\) contains 1 red marble and 9 green marbles only
Bag \(B\) contains 1 red marble and 4 green marbles only
Bag \(C\) contains 1 red marble and 2 green marbles only
Sasha selects at random one marble from bag \(A\).
If he selects a red marble, he stops selecting.
If the marble is green, he continues by selecting at random one marble from bag \(B\).
If he selects a red marble, he stops selecting.
If the marble is green, he continues by selecting at random one marble from bag \(C\).

1. Draw a tree diagram to represent this information.
2. Find the probability that Sasha selects 3 green marbles.
3. Find the probability that Sasha selects at least 1 marble of each colour.
4. Given that Sasha selects a red marble, find the probability that he selects it from bag \(B\).

1. Sally plays a game in which she can either win or lose.

A turn consists of up to 3 games. On each turn Sally plays the game up to 3 times. If she wins the first 2 games or loses the first 2 games, then she will not play the 3rd game.

• The probability that Sally wins the first game in a turn is 0.7
• If Sally wins a game the probability that she wins the next game is 0.6
• If Sally loses a game the probability that she wins the next game is 0.2
  1. Use this information to complete the tree diagram on page 3
  2. Find the probability that Sally wins the first 2 games in a turn.
  3. Find the probability that Sally wins exactly 2 games in a turn.

Given that Sally wins 2 games in a turn,

find the probability that she won the first 2 games. Given that Sally won the first game in a turn,

find the probability that she won 2 games. 1st game 2nd game win

6 Answer part (i) of this question on the insert provided. Mancaster Hockey Club invite prospective new players to take part in a series of three trial games. At the end of each game the performance of each player is assessed as pass or fail. Players who achieve a pass in all three games are invited to join the first team squad. Players who achieve a pass in two games are invited to join the second team squad. Players who fail in two games are asked to leave. This may happen after two games.

• The probability of passing the first game is 0.9
• Players who pass any game have probability 0.9 of passing the next game
• Players who fail any game have probability 0.5 of failing the next game
  1. On the insert, complete the tree diagram which illustrates the information above.
     \includegraphics[max width=\textwidth, alt={}, center]{668963b4-994d-475a-a1c8-c3e3a252e4e6-4_691_1329_978_397}
  2. Find the probability that a randomly selected player
     (A) is invited to join the first team squad,
     (B) is invited to join the second team squad.
  3. Hence write down the probability that a randomly selected player is asked to leave.
  4. Find the probability that a randomly selected player is asked to leave after two games, given that the player is asked to leave.

Angela, Bryony and Shareen attend the trials at the same time. Assuming their performances are independent, find the probability that

at least one of the three is asked to leave,

they pass a total of 7 games between them.

4 Answer part (i) of this question on the insert provided. Mancaster Hockey Club invite prospective new players to take part in a series of three trial games. At the end of each game the performance of each player is assessed as pass or fail. Players who achieve a pass in all three games are invited to join the first team squad. Players who achieve a pass in two games are invited to join the second team squad. Players who fail in two games are asked to leave. This may happen after two games.

• The probability of passing the first game is 0.9
• Players who pass any game have probability 0.9 of passing the next game
• Players who fail any game have probability 0.5 of failing the next game
  1. On the insert, complete the tree diagram which illustrates the information above.
     \includegraphics[max width=\textwidth, alt={}, center]{64f25a40-d3bf-4212-b92e-655f980c702b-4_643_1239_942_417}
  2. Find the probability that a randomly selected player
     (A) is invited to join the first team squad,
     (B) is invited to join the second team squad.
  3. Hence write down the probability that a randomly selected player is asked to leave.
  4. Find the probability that a randomly selected player is asked to leave after two games, given that the player is asked to leave.

Angela, Bryony and Shareen attend the trials at the same time. Assuming their performances are independent, find the probability that

at least one of the three is asked to leave,

they pass a total of 7 games between them.

1. Three Bags, \(A , B\) and \(C\), each contain 1 red marble and some green marbles.

\begin{displayquote} Bag \(A\) contains 1 red marble and 9 green marbles only
Bag \(B\) contains 1 red marble and 4 green marbles only
Bag \(C\) contains 1 red marble and 2 green marbles only \end{displayquote} Sasha selects at random one marble from \(\operatorname { Bag } A\).
If he selects a red marble, he stops selecting.
If the marble is green, he continues by selecting at random one marble from Bag \(B\).
If he selects a red marble, he stops selecting.
If the marble is green, he continues by selecting at random one marble from Bag \(C\).

1. Draw a tree diagram to represent this information.
2. Find the probability that Sasha selects 3 green marbles.
3. Find the probability that Sasha selects at least 1 marble of each colour.
4. Given that Sasha selects a red marble, find the probability that he selects it from Bag \(B\).