2.03c Conditional probability: using diagrams/tables

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]

1. The table below shows the destination from school of 180 year 11 pupils. Most pupils either continued education, in school or college, or went into some form of employment.

A reporter selects two pupils at random to interview. Given that the first pupil is in school or college, find the probability that both pupils are girls.

12 You must show detailed reasoning in this question. In the summer of 2017 in England a large number of candidates sat GCSE examinations in both mathematics and English. 56\% of these candidates achieved at least level 4 in mathematics and \(80 \%\) of these candidates achieved at least level 4 in English. 14\% of these candidates did not achieve at least level 4 in either mathematics or English. Determine whether achieving level 4 or above in English and achieving level 4 or above in mathematics were independent events.

12 Rob has two six-sided dice, each with sides numbered 1, 2, 3, 4, 5, 6.
One dice is fair. The other dice is biased, with probabilities as shown in the table. Rob throws each dice once and notes the two scores, \(X\) on the fair dice and \(Y\) on the biased dice. He then calculates the value of the variable \(S\) which is defined as follows.

• If \(X \leqslant 3\), then \(S = X + 2 Y\).
• If \(X > 3\), then \(S = X + Y\).
  1. (a) Draw up a sample space diagram showing all the possible outcomes and the corresponding values of \(S\).
     (b) On your diagram, circle the four cells where the value \(S = 10\) occurs.
  2. Explain the mistake in the following calculation.

$$\mathrm { P } ( S = 10 ) = \frac { \text { Number of outcomes giving } S = 10 } { \text { Total number of outcomes } } = \frac { 4 } { 36 } = \frac { 1 } { 9 } .$$

Find the correct value of \(\mathrm { P } ( S = 10 )\).

Given that \(S = 10\), find the probability that the score on one of the dice is 4 .

The events " \(X = 1\) or 2 " and " \(S = n\) " are mutually exclusive. Given that \(\mathrm { P } ( S = n ) \neq 0\), find the value of \(n\).

1 Each of the 180 students at a college plays exactly one of the piano, the guitar and the drums. The numbers of male and female students who play the piano, the guitar and the drums are given in the following table. A student at the college is chosen at random.

1. Find the probability that the student plays the guitar.
2. Find the probability that the student is male given that the student plays the drums.
3. Determine whether the events 'the student plays the guitar' and 'the student is female' are independent, justifying your answer.

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 The table below shows the numbers of males and females in each of three employment categories at a university on 31 July 2003.

1. An employee is selected at random. Determine the probability that the employee is:
   1. female;
   2. a female academic;
   3. either female or academic or both;
   4. female, given that the employee is academic.
2. Three employees are selected at random, without replacement. Determine the probability that:
   1. all three employees are male;
   2. exactly one employee is male.
3. The event "employee selected is academic" is denoted by \(A\). The event "employee selected is female" is denoted by \(F\). Describe in context, as simply as possible, the events denoted by:
   1. \(F \cap A\);
   2. \(F ^ { \prime } \cup A\).

      Surname	Other Names
      Centre Number				Candidate Number
      Candidate Signature

      General Certificate of Education
      January 2005
      Advanced Subsidiary Examination MS/SS1B AQA
      459:5EMLM
      : 11 P וPII " 1 : : ר
      ALLI.ub c \section*{STATISTICS} Unit Statistics 1B Insert for use in Question 3.
      Fill in the boxes at the top of this page.
      Fasten this insert securely to your answer book. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Scatter diagram for parcel deliveries by a van} \includegraphics[alt={},max width=\textwidth]{7faa4a2d-f5cc-4cc3-a3a9-5d8290ceabdc-8_2420_1664_349_175}
      \end{figure} Figure 1 (for Question 3)

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.

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}

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]
   1. \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}

6 Two bags contain coloured discs. At first, bag \(P\) contains 2 red discs and 2 green discs, and bag \(Q\) contains 3 red discs and 1 green disc. A disc is chosen at random from bag \(P\), its colour is noted and it is placed in bag \(Q\). A disc is then chosen at random from bag \(Q\), its colour is noted and it is placed in bag \(P\). A disc is then chosen at random from bag \(P\). The tree diagram shows the different combinations of three coloured discs chosen. \includegraphics[max width=\textwidth, alt={}, center]{11316ea6-3999-4003-b77d-bee8b547c1da-05_858_980_573_585}

1. Write down the values of \(a , b , c , d , e\) and \(f\). The total number of red discs chosen, out of 3, is denoted by \(R\). The table shows the probability distribution of \(R\).

   \(r\)	0	1	2	3
   \(\mathrm { P } ( R = r )\)	\(\frac { 1 } { 10 }\)	\(k\)	\(\frac { 9 } { 20 }\)	\(\frac { 1 } { 5 }\)

2. Show how to obtain the value \(\mathrm { P } ( R = 2 ) = \frac { 9 } { 20 }\).
3. Find the value of \(k\).
4. Calculate the mean and variance of \(R\).

3 Each enquiry received by a business support unit is dealt with by Ewan, Fay or Gaby. The probabilities of them dealing with an enquiry are \(0.2,0.3\) and 0.5 respectively. Of enquiries dealt with by Ewan, 60\% are answered immediately, 25\% are answered later the same day and the remainder are answered at a later date. Of enquiries dealt with by Fay, 75\% are answered immediately, 15\% are answered later the same day and the remainder are answered at a later date. Of enquiries dealt with by Gaby, 90\% are answered immediately and the remainder are answered at a later date.

1. Determine the probability that an enquiry:
   1. is dealt with by Gaby and answered immediately;
   2. is answered immediately;
   3. is dealt with by Gaby, given that it is answered immediately.
2. Determine the probability that an enquiry is dealt with by Ewan, given that it is answered later the same day.

2 A hill-top monument can be visited by one of three routes: road, funicular railway or cable car. The percentages of visitors using these routes are 25, 35 and 40 respectively. The age distribution, in percentages, of visitors using each route is shown in the table. For example, 15 per cent of visitors using the road were under 18 . Calculate the probability that a randomly selected visitor:

1. who used the road is aged 18 or over;
2. is aged between 18 and 64;
3. used the funicular railway and is aged over 64;
4. used the funicular railway, given that the visitor is aged over 64.

13 There are 25 students in a class.

• The number of students who study both History and English is 3.
• The number of students who study neither History nor English is 14 .
• The number of students who study History but not English is three times the number who study English but not History.
  1. - Show this information on a Venn diagram.
  2. Determine the probability that a student selected at random studies English.

Two different students from the class are chosen at random.

Given that exactly one of the two students studies English, determine the probability that exactly one of the two students studies History. \section*{END OF QUESTION PAPER}

1. The Venn diagram shows the probabilities for students at a college taking part in various sports. \(A\) represents the event that a student takes part in Athletics. \(T\) represents the event that a student takes part in Tennis. \(C\) represents the event that a student takes part in Cricket. \(p\) and \(q\) are probabilities. \includegraphics[max width=\textwidth, alt={}, center]{8f3dbcb4-3260-4493-a230-12577b4ed691-06_668_935_596_566}

The probability that a student selected at random takes part in Athletics or Tennis is 0.75

1. Find the value of \(p\).
2. State, giving a reason, whether or not the events \(A\) and \(T\) are statistically independent. Show your working clearly.
3. Find the probability that a student selected at random does not take part in Athletics or Cricket.

1. Given that

$$\mathrm { P } ( A ) = 0.35 \quad \mathrm { P } ( B ) = 0.45 \quad \text { and } \quad \mathrm { P } ( A \cap B ) = 0.13$$ find

1. \(\mathrm { P } \left( A ^ { \prime } \mid B ^ { \prime } \right)\)
2. Explain why the events \(A\) and \(B\) are not independent. The event \(C\) has \(\mathrm { P } ( C ) = 0.20\) The events \(A\) and \(C\) are mutually exclusive and the events \(B\) and \(C\) are statistically independent.
3. Draw a Venn diagram to illustrate the events \(A , B\) and \(C\), giving the probabilities for each region.
4. Find \(\mathrm { P } \left( [ B \cup C ] ^ { \prime } \right)\)

4. The Venn diagram shows the probabilities of students' lunch boxes containing a drink, sandwiches and a chocolate bar. \includegraphics[max width=\textwidth, alt={}, center]{565bfa73-8095-4242-80b6-cd47aaff6a31-05_655_899_392_484} \(D\) is the event that a lunch box contains a drink, \(S\) is the event that a lunch box contains sandwiches, \(C\) is the event that a lunch box contains a chocolate bar, \(u , v\) and \(w\) are probabilities.

1. Write down \(\mathrm { P } \left( S \cap D ^ { \prime } \right)\). One day, 80 students each bring in a lunch box.
   Given that all 80 lunch boxes contain sandwiches and a drink,
2. estimate how many of these 80 lunch boxes will contain a chocolate bar. Given that the events \(S\) and \(C\) are independent and that \(\mathrm { P } ( D \mid C ) = \frac { 14 } { 15 }\),
3. calculate the value of \(u\), the value of \(v\) and the value of \(w\).
   (7)
   (Total 11 marks)