Sequential events and tree diagrams

5 Three friends, Rick, Brenda and Ali, go to a football match but forget to say which entrance to the ground they will meet at. There are four entrances, \(A , B , C\) and \(D\). Each friend chooses an entrance independently.

• The probability that Rick chooses entrance \(A\) is \(\frac { 1 } { 3 }\). The probabilities that he chooses entrances \(B , C\) or \(D\) are all equal.
• Brenda is equally likely to choose any of the four entrances.
• The probability that Ali chooses entrance \(C\) is \(\frac { 2 } { 7 }\) and the probability that he chooses entrance \(D\) is \(\frac { 3 } { 5 }\). The probabilities that he chooses the other two entrances are equal.
  1. Find the probability that at least 2 friends will choose entrance \(B\).
  2. Find the probability that the three friends will all choose the same entrance.

7 James has a fair coin and a fair tetrahedral die with four faces numbered 1, 2, 3, 4. He tosses the coin once and the die twice. The random variable \(X\) is defined as follows.

• If the coin shows a head then \(X\) is the sum of the scores on the two throws of the die.
• If the coin shows a tail then \(X\) is the score on the first throw of the die only.
  1. Explain why \(X = 1\) can only be obtained by throwing a tail, and show that \(\mathrm { P } ( X = 1 ) = \frac { 1 } { 8 }\).
  2. Show that \(\mathrm { P } ( X = 3 ) = \frac { 3 } { 16 }\).
  3. Copy and complete the probability distribution table for \(X\).

Event \(Q\) is 'James throws a tail'. Event \(R\) is 'the value of \(X\) is 7'.

Determine whether events \(Q\) and \(R\) are exclusive. Justify your answer.

8 Ann, Bill, Chris and Dipak play a game with a fair cubical die. Starting with Ann they take turns, in alphabetical order, to throw the die. This process is repeated as many times as necessary until a player throws a 6 . When this happens, the game stops and this player is the winner. Find the probability that

1. Chris wins on his first throw,
2. Dipak wins on his second throw,
3. Ann gets a third throw,
4. Bill throws the die exactly three times.

6. A software company sets exams for programmers who wish to qualify to use their packages. Past records show that \(55 \%\) of candidates taking the exam for the first time will pass, \(60 \%\) of those taking it for the second time will pass, but only \(40 \%\) of those taking the exam for the third time will pass. Candidates are not allowed to sit the exam more than three times. A programmer decides to keep taking the exam until he passes or is allowed no further attempts. Find the probability that he will

1. pass the exam on his second attempt,
2. pass the exam. Another programmer already has the qualification.
3. Find, correct to 3 significant figures, the probability that she passed first time. At a particular sitting of the exam there are 400 candidates.
   The ratio of those sitting the exam for the first time to those sitting it for the second time to those sitting it for the third time is \(5 : 3 : 2\)
4. How many of the 400 candidates would be expected to pass?

2. A factory buys \(10 \%\) of its components from supplier \(A , 30 \%\) from supplier \(B\) and the rest from supplier \(C\). It is known that \(6 \%\) of the components it buys are faulty. Of the components bought from supplier \(A , 9 \%\) are faulty and of the components bought from supplier \(B , 3 \%\) are faulty.

1. Find the percentage of components bought from supplier \(C\) that are faulty. A component is selected at random.
2. Explain why the event "the component was bought from supplier \(B\) " is not statistically independent from the event "the component is faulty".
   [0pt] [BLANK PAGE]

14 Mr Jones has 3 tins of beans and 2 tins of pears. His daughter has removed the labels for a school project, and the tins are identical in appearance. Mr Jones opens tins in turn until he has opened at least 1 tin of beans and at least 1 tin of pears. He does not open any remaining tins.

1. Draw a tree diagram to illustrate this situation, labelling each branch with its associated probability.
2. Find the probability that Mr Jones opens exactly 3 tins.
3. It is given that the last tin Mr Jones opens is a tin of pears. Find the probability that he opens exactly 3 tins.

3 Fred and his daughter, Delia, support their town's rugby team. The probability that Fred watches a game is 0.8 . The probability that Delia watches a game is 0.9 when her father watches the game, and is 0.4 when her father does not watch the game.

1. Calculate the probability that:
   1. both Fred and Delia watch a particular game;
   2. neither Fred nor Delia watch a particular game.
2. Molly supports the same rugby team as Fred and Delia. The probability that Molly watches a game is 0.7 , and is independent of whether or not Fred or Delia watches the game. Calculate the probability that:
   1. all 3 supporters watch a particular game;
   2. exactly 2 of the 3 supporters watch a particular game.

3 A ferry sails once each day from port D to port A. The ferry departs from D on time or late but never early. However, the ferry can arrive at A early, on time or late. The probabilities for some combined events of departing from \(D\) and arriving at \(A\) are shown in the table below.

1. Complete the table.
2. Write down the probability that, on a particular day, the ferry:
   1. both departs and arrives on time;
   2. departs late.
3. Find the probability that, on a particular day, the ferry:
   1. arrives late, given that it departed late;
   2. does not arrive late, given that it departed on time.
4. On three particular days, the ferry departs from port D on time. Find the probability that, on these three days, the ferry arrives at port A early once, on time once and late once. Give your answer to three decimal places.
   [0pt] [4 marks]
5. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Answer space for question 3}

   \multirow{2}{*}{}		Arrive at A
   		Early	On time	Late	Total
   \multirow{2}{*}{Depart from D}	On time	0.16	0.56	0.08
   	Late
   	Total	0.22	0.65		1.00

   \end{table}