2.03b Probability diagrams: tree, Venn, sample space

5. A College employs 75 teachers, of whom 47 are full-time and the rest are part-time. Of the 39 male teachers at the College, 26 are full-time.

1. Represent this information on a Venn diagram.
2. One teacher is selected at random to be interviewed by an inspector. Find the probability that the teacher chosen
   1. works full-time and is female,
   2. works part-time, given that he is male.
3. Three teachers are selected at random to be observed by an inspector during one day. Find correct to 3 significant figures the probability that
   1. all three teachers chosen work full-time,
   2. at least one of the three teachers chosen is female.

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)

4 A manufacturer produces three models of washing machine: basic, standard and deluxe. An analysis of warranty records shows that \(25 \%\) of faults are on basic machines, \(60 \%\) are on standard machines and 15\% are on deluxe machines. For basic machines, 30\% of faults reported during the warranty period are electrical, \(50 \%\) are mechanical and \(20 \%\) are water-related. For standard machines, 40\% of faults reported during the warranty period are electrical, \(45 \%\) are mechanical and 15\% are water-related. For deluxe machines, \(55 \%\) of faults reported during the warranty period are electrical, \(35 \%\) are mechanical and \(10 \%\) are water-related.

1. Draw a tree diagram to represent the above information.
2. Hence, or otherwise, determine the probability that a fault reported during the warranty period:
   1. is electrical;
   2. is on a deluxe machine, given that it is electrical.
3. A random sample of 10 electrical faults reported during the warranty period is selected. Calculate the probability that exactly 4 of them are on deluxe machines.

2 A hotel chain has hotels in three types of location: city, coastal and country. The percentages of the chain's reservations for each of these locations are 30,55 and 15 respectively. Each of the chain's hotels offers three types of reservation: Bed \& Breakfast, Half Board and Full Board. The percentages of these types of reservation for each of the three types of location are shown in the table. For example, 80 per cent of reservations for hotels in city locations are for Bed \& Breakfast.

1. For a reservation selected at random:
   1. show that the probability that it is for Bed \& Breakfast is 0.34 ;
   2. calculate the probability that it is for Half Board in a hotel in a coastal location;
   3. calculate the probability that it is for a hotel in a coastal location, given that it is for Half Board.
2. A random sample of 3 reservations for Half Board is selected. Calculate the probability that these 3 reservations are for hotels in different types of location.

4 It is proposed to introduce, for all males at age 60, screening tests, A and B, for a certain disease. Test B is administered only when the result of Test A is inconclusive. It is known that 10\% of 60-year-old men suffer from the disease. For those 60 -year-old men suffering from the disease:

• Test A is known to give a positive result, indicating a presence of the disease, in \(90 \%\) of cases, a negative result in \(2 \%\) of cases and a requirement for the administration of Test B in \(8 \%\) of cases;
• Test B is known to give a positive result in \(98 \%\) of cases and a negative result in 2\% of cases.

For those 60 -year-old men not suffering from the disease:

• Test A is known to give a positive result in \(1 \%\) of cases, a negative result in \(80 \%\) of cases and a requirement for the administration of Test B in 19\% of cases;
• Test B is known to give a positive result in \(1 \%\) of cases and a negative result in \(99 \%\) of cases.

3 An IT help desk has three telephone stations: Alpha, Beta and Gamma. Each of these stations deals only with telephone enquiries. The probability that an enquiry is received at Alpha is 0.60 .
The probability that an enquiry is received at Beta is 0.25 .
The probability that an enquiry is received at Gamma is 0.15 . Each enquiry is resolved at the station that receives the enquiry. The percentages of enquiries resolved within various times at each station are shown in the table. For example:
80 per cent of enquiries received at Alpha are resolved within 24 hours;
25 per cent of enquiries received at Alpha take between 1 hour and 24 hours to resolve.

1. Find the probability that an enquiry, selected at random, is:
   1. resolved at Gamma;
   2. resolved at Alpha within 1 hour;
   3. resolved within 24 hours;
   4. received at Beta, given that it is resolved within 24 hours.
2. A random sample of 3 enquiries was selected. Given that all 3 enquiries were resolved within 24 hours, calculate the probability that they were all received at:
   1. Beta;
   2. the same station.
      

3 A hotel has three types of room: double, twin and suite. The percentage of rooms in the hotel of each type is 40,45 and 15 respectively. Each room in the hotel may be occupied by \(0,1,2\), or 3 or more people. The proportional occupancy of each type of room is shown in the table.

2 On a rail route between two stations, A and \(\mathrm { B } , 90 \%\) of trains leave A on time and \(10 \%\) of trains leave A late. Of those trains that leave A on time, \(15 \%\) arrive at B early, \(75 \%\) arrive on time and \(10 \%\) arrive late. Of those trains that leave A late, \(35 \%\) arrive at B on time and \(65 \%\) arrive late.

1. Represent this information by a fully-labelled tree diagram.
2. Hence, or otherwise, calculate the probability that a train:
   1. arrives at B early or on time;
   2. left A on time, given that it arrived at B on time;
   3. left A late, given that it was not late in arriving at B .
3. Two trains arrive late at B. Assuming that their journey times are independent, calculate the probability that exactly one train left A on time.

3 An investigation was carried out into the type of vehicle being driven when its driver was caught speeding. The investigation was restricted to drivers who were caught speeding when driving vehicles with at least 4 wheels. An analysis of the results showed that \(65 \%\) were driving cars ( C ), \(20 \%\) were driving vans (V) and 15\% were driving lorries (L). Of those driving cars, \(30 \%\) were caught by fixed speed cameras (F), 55\% were caught by mobile speed cameras (M) and 15\% were caught by average speed cameras (A). Of those driving vans, \(35 \%\) were caught by fixed speed cameras (F), \(45 \%\) were caught by mobile speed cameras (M) and 20\% were caught by average speed cameras (A). Of those driving lorries, \(10 \%\) were caught by fixed speed cameras \(( \mathrm { F } )\), \(65 \%\) were caught by mobile speed cameras (M) and \(25 \%\) were caught by average speed cameras (A).

1. Represent this information by a tree diagram on which are shown labels and percentages or probabilities.
2. Hence, or otherwise, calculate the probability that a driver, selected at random from those caught speeding:
   1. was driving either a car or a lorry and was caught by a mobile speed camera;
   2. was driving a lorry, given that the driver was caught by an average speed camera;
   3. was not caught by a fixed speed camera, given that the driver was not driving a car.
      [0pt] [8 marks]
3. Three drivers were selected at random from those caught speeding by fixed speed cameras. Calculate the probability that they were driving three different types of vehicle.
   [0pt] [4 marks]

3 A particular brand of spread is produced in three varieties: standard, light and very light. During a marketing campaign, the producer advertises that some cartons of spread contain coupons worth \(\pounds 1 , \pounds 2\) or \(\pounds 4\). For each variety of spread, the proportion of cartons containing coupons of each value is shown in the table. For example, the probability that a carton of standard spread contains a coupon worth \(\pounds 2\) is 0.08 . In a large batch of cartons, 55 per cent contain standard spread, 30 per cent contain light spread and 15 per cent contain very light spread.

1. A carton of spread is selected at random from the batch. Find the probability that the carton:
   1. contains standard spread and a coupon worth \(\pounds 1\);
   2. does not contain a coupon;
   3. contains light spread, given that it does not contain a coupon;
   4. contains very light spread, given that it contains a coupon.
2. A random sample of 3 cartons is selected from the batch. Given that all of these 3 cartons contain a coupon, find the probability that they each contain a different variety of spread.
   [0pt] [4 marks]

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.

13 Bag A contains 3 black discs and 2 white discs only. Initially Bag B is empty. Discs are removed at random from bag A, and are placed in bag B, one at a time, until all 5 discs are in bag B.

1. Write down the probability that the last disc that is placed in bag B is black.
2. Find the probability that the first disc and the last disc that are placed in bag B are both black.
3. Find the probability that, starting from when the first disc is placed in bag B , the number of black discs in bag B is always greater than the number of white discs in bag B.

1. A factory produces shoes.

A quality control inspector at the factory checks a sample of 120 shoes for each of three types of defect. The Venn diagram represents the inspector's results. A represents the event that a shoe has defective stitching \(B\) represents the event that a shoe has defective colouring \(C\) represents the event that a shoe has defective soles \includegraphics[max width=\textwidth, alt={}, center]{fa1cb8a2-dab9-4133-b7a1-9108888c37d7-02_684_935_607_566} One of the shoes in the sample is selected at random.

1. Find the probability that it does not have defective soles.
2. Find \(\mathrm { P } \left( A \cap B \cap C ^ { \prime } \right)\)
3. Find \(\mathrm { P } \left( A \cup B \cup C ^ { \prime } \right)\)
4. Find the probability that the shoe has at most one type of defect.
5. Given the selected shoe has at most one type of defect, find the probability it has defective stitching. The random variable \(X\) is the number of the events \(A , B , C\) that occur for a randomly selected shoe.
6. Find \(\mathrm { E } ( X )\) \section*{This is a copy of the Venn diagram for this question.} \includegraphics[max width=\textwidth, alt={}, center]{fa1cb8a2-dab9-4133-b7a1-9108888c37d7-05_684_940_388_566}

4.The partially completed tree diagram,where \(p\) and \(q\) are probabilities,gives information about Andrew's journey to work each day. \includegraphics[max width=\textwidth, alt={}, center]{7d45bacd-20ac-49b4-8f3f-613edf3739f9-12_661_794_395_511} \(R\) represents the event that it is raining
W represents the event that Andrew walks to work \(B\) represents the event that Andrew takes the bus to work \(C\) represents the event that Andrew cycles to work Given that \(\mathrm { P } ( B ) = 0.26\)

1. find the value of \(p\) Given also that \(\mathrm { P } \left( R ^ { \prime } \mid W \right) = 0.175\)
2. find the value of \(q\)
3. Find the probability that Andrew cycles to work. Given that Andrew did not cycle to work on Friday,
4. find the probability that it was raining on Friday.

1. The Venn diagram, where \(w , x , y\) and \(z\) are probabilities, shows the probabilities of a group of students buying each of 3 magazines.

A represents the event that a student buys magazine \(A\) and \(\mathrm { P } ( A ) = 0.60\) \(B\) represents the event that a student buys magazine \(B\) and \(\mathrm { P } ( B ) = 0.15\) \(C\) represents the event that a student buys magazine \(C\) and \(\mathrm { P } ( C ) = 0.35\) \includegraphics[max width=\textwidth, alt={}, center]{77ae01cd-2b58-48ab-889f-272e27ecf99d-06_504_755_641_596}

1. State which two of the three events \(A\), \(B\) and \(C\) are mutually exclusive. The events \(A\) and \(C\) are independent.
2. Show that \(w = 0.21\)
3. Find the value of \(x\), the value of \(y\) and the value of \(z\).
4. Find the probability that a student selected at random buys only one of these magazines.
5. Find the probability that a student selected at random buys magazine \(B\) or magazine \(C\).
6. Find \(\mathrm { P } ( A \mid [ B \cup C ] )\)

1. The following incomplete tree diagram shows the relationships between the event \(A\) and the event \(B\). \includegraphics[max width=\textwidth, alt={}, center]{77ae01cd-2b58-48ab-889f-272e27ecf99d-14_799_839_351_548}

Given that \(\mathrm { P } ( B ) = \frac { 9 } { 20 }\)

1. find \(\mathrm { P } ( A )\) and complete the tree diagram,
2. find \(\mathrm { P } \left( A ^ { \prime } \mid B ^ { \prime } \right)\).

4. Three bags A, B and \(\mathbf { C }\) each contain coloured balls. Bag A contains 4 red balls and 2 yellow balls only.
Bag B contains 4 red balls and 1 yellow ball only.
Bag \(\mathbf { C }\) contains 6 red balls only. In a game
Mike takes a ball at random from bag \(\mathbf { A }\), records the colour and places it in bag \(\mathbf { C }\). He then takes a ball at random from bag \(\mathbf { B }\), records the colour and places it in bag \(\mathbf { C }\). Finally, Mike takes a ball at random from bag \(\mathbf { C }\) and records the colour.

1. Complete the tree diagram on the page opposite, to illustrate the game by adding the remaining branches and all probabilities.
2. Show that the probability that Mike records a yellow ball exactly twice is \(\frac { 1 } { 10 }\) Given that Mike records exactly 2 yellow balls,
3. find the probability that the ball drawn from bag \(\mathbf { A }\) is red. Mike plays this game a large number of times, each time starting with the bags containing balls as described above. The random variable \(X\) represents the number of yellow balls recorded in a single game.
4. Find the probability distribution of \(X\)
5. Find \(\mathrm { E } ( X )\) Bag B
   Bag C \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Bag A} \includegraphics[alt={},max width=\textwidth]{29ac0c0b-f963-40a1-beba-7146bbb2d021-13_739_1580_411_182}
   \end{figure}

4. The employees of a company are classified as management, administration or production. The following table shows the number employed in each category and whether or not they live close to the company or some distance away. An employee is chosen at random.
Find the probability that this employee

1. is an administrator,
2. lives close to the company, given that the employee is a manager. Of the managers, \(90 \%\) are married, as are \(60 \%\) of the administrators and \(80 \%\) of the production employees.
3. Construct a tree diagram containing all the probabilities.
4. Find the probability that an employee chosen at random is married. (3 marks) An employee is selected at random and found to be married.
5. Find the probability that this employee is in production.

7. In a school there are 148 students in Years 12 and 13 studying Science, Humanities or Arts subjects. Of these students, 89 wear glasses and the others do not. There are 30 Science students of whom 18 wear glasses. The corresponding figures for the Humanities students are 68 and 44 respectively. A student is chosen at random. Find the probability that this student

1. is studying Arts subjects,
2. does not wear glasses, given that the student is studying Arts subjects. Amongst the Science students, \(80 \%\) are right-handed. Corresponding percentages for Humanities and Arts students are 75\% and 70\% respectively. A student is again chosen at random.
3. Find the probability that this student is right-handed.
4. Given that this student is right-handed, find the probability that the student is studying Science subjects.
   1. (a) Describe the main features and uses of a box plot.
      
   Children from schools \(A\) and \(B\) took part in a fun run for charity. The times, to the nearest minute, taken by the children from school \(A\) are summarised in Figure 1. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{3d4f7bfb-b235-418a-9411-a4d0b3188254-015_398_1045_946_461}
   \end{figure}
5. 1. Write down the time by which \(75 \%\) of the children in school \(A\) had completed the run.
   2. State the name given to this value.
6. Explain what you understand by the two crosses ( X ) on Figure 1.

6 A housing estate consists of 320 houses: 120 detached and 200 semi-detached. The numbers of children living in these houses are shown in the table. A house on the estate is selected at random. \(D\) denotes the event 'the house is detached'. \(R\) denotes the event 'no children live in the house'. \(S\) denotes the event 'one child lives in the house'. \(T\) denotes the event 'two children live in the house'.
( \(D ^ { \prime }\) denotes the event 'not \(D\) '.)

1. Find:
   1. \(\mathrm { P } ( D )\);
   2. \(\quad \mathrm { P } ( D \cap R )\);
   3. \(\quad \mathrm { P } ( D \cup T )\);
   4. \(\mathrm { P } ( D \mid R )\);
   5. \(\mathrm { P } \left( R \mid D ^ { \prime } \right)\).
2. 1. Name two of the events \(D , R , S\) and \(T\) that are mutually exclusive.
   2. Determine whether the events \(D\) and \(R\) are independent. Justify your answer.
3. Define, in the context of this question, the event:
   1. \(D ^ { \prime } \cup T\);
   2. \(D \cap ( R \cup S )\).

4

1. Chris shops at his local store on his way to and from work every Friday.
   The event that he buys a morning newspaper is denoted by \(M\), and the event that he buys an evening newspaper is denoted by \(E\). On any one Friday, Chris may buy neither, exactly one or both of these newspapers.
   1. Complete the table of probabilities, printed on the opposite page, where \(M ^ { \prime }\) and \(E ^ { \prime }\) denote the events 'not \(M\) ' and 'not \(E\) ' respectively.
   2. Hence, or otherwise, find the probability that, on any given Friday, Chris buys exactly one newspaper.
   3. Give a numerical justification for the following statement.
      'The events \(M\) and \(E\) are not mutually exclusive.'
2. The event that Chris buys a morning newspaper on Saturday is denoted by \(S\), and the event that he buys a morning newspaper on the following day, Sunday, is denoted by \(T\). The event that he buys a morning newspaper on both Saturday and Sunday is denoted by \(S \cap T\). Each combination of the events \(S\) and \(T\) is independent of any combination of the events \(M\) and \(E\). However, the events \(S\) and \(T\) are not independent, with $$\mathrm { P } ( S ) = 0.85 , \quad \mathrm { P } ( T \mid S ) = 0.20 \quad \text { and } \quad \mathrm { P } \left( T \mid S ^ { \prime } \right) = 0.75$$ Find the probability that, on a particular Friday, Saturday and Sunday, Chris buys:
   1. all four newspapers;
   2. none of the four newspapers.
3. 1. State, as briefly as possible, in the context of the question, the event that is denoted by \(M \cap E ^ { \prime } \cap S \cap T ^ { \prime }\).
   2. Calculate the value of \(\mathrm { P } \left( M \cap E ^ { \prime } \cap S \cap T ^ { \prime } \right)\). \section*{Answer space for question 4}
      1. (i)

         \cline { 2 - 4 } \multicolumn{1}{c|}{}	\(\boldsymbol { M }\)	\(\boldsymbol { M } ^ { \prime }\)	Total
         \(\boldsymbol { E }\)	0.16		0.28
         \(\boldsymbol { E } ^ { \prime }\)
         Total		0.60	1.00

         \includegraphics[max width=\textwidth, alt={}]{4c679380-894f-4d36-aec8-296b662058e2-11_2050_1707_687_153}