Repeated trials until stopping condition

5 A game is played with an ordinary fair 6-sided die. A player throws the die once. If the result is \(2,3,4\) or 5 , that result is the player's score and the player does not throw the die again. If the result is 1 or 6 , the player throws the die a second time and the player's score is the sum of the two numbers from the two throws.

1. Draw a fully labelled tree diagram to represent this information. Events \(A\) and \(B\) are defined as follows.
   \(A\) : the player's score is \(5,6,7,8\) or 9
   \(B\) : the player has two throws
2. Show that \(\mathrm { P } ( A ) = \frac { 1 } { 3 }\).
3. Determine whether or not events \(A\) and \(B\) are independent.
4. Find \(\mathrm { P } \left( B \mid A ^ { \prime } \right)\).

6 When Don plays tennis, \(65 \%\) of his first serves go into the correct area of the court. If the first serve goes into the correct area, his chance of winning the point is \(90 \%\). If his first serve does not go into the correct area, Don is allowed a second serve, and of these, \(80 \%\) go into the correct area. If the second serve goes into the correct area, his chance of winning the point is \(60 \%\). If neither serve goes into the correct area, Don loses the point.

1. Draw a tree diagram to represent this information.
2. Using your tree diagram, find the probability that Don loses the point.
3. Find the conditional probability that Don's first serve went into the correct area, given that he loses the point.

8 A game is played with a fair, six-sided die which has 4 red faces and 2 blue faces. One turn consists of throwing the die repeatedly until a blue face is on top or until the die has been thrown 4 times.

1. In the answer book, complete the probability tree diagram for one turn.
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2. Find the probability that in one particular turn the die is thrown 4 times.
3. Adnan and Beryl each have one turn. Find the probability that Adnan throws the die more times than Beryl.
4. State one change that needs to be made to the rules so that the number of throws in one turn will have a geometric distribution.

2 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.

12 Anika and Beth are playing a game which consists of several points.

• The probability that Anika will win any point is 0.7 .
• The probability that Beth will win any point is 0.3 .
• The outcome of each point is independent of the outcome of every other point.

The first player to win two points wins the game.

1. Write down the probability that the game consists of more than three points.
2. Complete the probability tree diagram in the Printed Answer Booklet showing all the possibilities for the game.
3. Determine the probability that Beth wins the game.
4. Determine the probability that the game consists of exactly three points.
5. Given that Beth wins the game, determine the probability that the game consists of exactly three points.