2.03a Mutually exclusive and independent events

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]

2 Leila and Caleb are playing a game, using fair six-sided dice and unbiased coins.

• Leila rolls two dice, and her score \(L\) is the total of the scores on the two dice.
• Caleb spins 4 coins and his score \(C\) is three times the number of heads obtained.

The winner of a game is the player with the higher score. If the two scores are equal, the result of the game is a draw. The spreadsheet in Fig. 2 shows a simulation of 20 plays of the game. \begin{table}[h] \end{table}

1. Explain why the value of \(C\) in row 2 is 3 .
2. Use the spreadsheet to estimate \(\mathrm { P } ( C > 6 )\) and \(\mathrm { P } ( L > 6 )\).
3. Use the spreadsheet to estimate the probability that Leila loses a randomly chosen game.
4. Explain why your answers to parts (b) and (c) may not be very close to the true values.
5. Leila claims that the game is fair (that Leila and Caleb each have an equal chance of winning) because both she and Caleb can get a maximum score of 12 and also in the simulation she won exactly \(50 \%\) of the games.
   Make 2 comments about Leila's claim.

2 Natasha is investigating binomial distributions. She constructs the spreadsheet in Fig. 2 which shows the first 3 and last 4 rows of a simulation involving two independent variables, \(X\) and \(Y\), with distributions \(\mathrm { B } ( 10,0.3 )\) and \(\mathrm { B } ( 50,0.3 )\) respectively. The spreadsheet also shows the corresponding value of the random variable \(Z\), defined by \(Z = 5 X - Y\), for each pair of values of \(X\) and \(Y\). There are 100 simulated values of each of \(X , Y\) and \(Z\). The spreadsheet also shows whether each value of \(Z\) is greater than 6, and cells D103 and D104 show the number of values of \(Z\) which are greater than 6 and not greater than 6 respectively. \begin{table}[h] \end{table}

1. Use the information in the spreadsheet to write down an estimate of \(\mathrm { P } ( Z > 6 )\).
2. Explain how a more reliable estimate of \(\mathrm { P } ( Z > 6 )\) could be obtained.
3. 1. State the greatest possible value of \(Z\).
   2. Explain why it is very unlikely that \(Z\) would have this value.
4. Use the Central Limit Theorem to calculate an estimate of the probability that the mean of 20 independent values of \(Z\) is greater than 2 .

1 Abby runs a stall at a charity event. Visitors to the stall pay to play a game in which six fair dice are rolled. If the difference between the highest and lowest scores is less than 3 then the player wins \(\pounds 5\). Otherwise the player wins nothing. Abby designs the spreadsheet shown in Fig. 1 to estimate the probability of a player winning, by simulating 20 goes at the game. Cell C5, highlighted, shows that the 2nd dice in simulated game 4 scores 5 . Cells H5 and I5 show the highest and lowest scores, respectively, in game 4, and cell J5 gives the difference between them. \begin{table}[h] \end{table}

1. (A) Write down the numbers in columns H , I and J for game 20 .
   (B) Use the spreadsheet to estimate the probability of a player winning a game.
2. State how the estimate of probability in (i) (B) could be improved.
3. Give one advantage and one disadvantage of using this simulation technique compared with working out the theoretical probability. All profit made by the stall is given to charity. Abby has to decide how much to charge players to play.
4. If Abby charges \(\pounds 1\) per game, estimate the total profit when 50 players each play the game once.

Jake works for a parcel delivery company. The masses, in kilograms, of parcels he delivers are normally distributed with mean \(2 \cdot 2\) and standard deviation \(0 \cdot 3\).

1. Calculate the probability that a randomly selected parcel will have a mass less than 1.8 kg .
   Jake delivers the lightest \(80 \%\) of parcels on his bike. The rest he puts in his car and delivers by car.
2. Find the mass of the heaviest parcel he would deliver by bike.
   
3. He randomly selects a parcel from his car. Find the probability that it has a mass less than 3 kg .
   
4. In the run-up to Christmas, Jake believes that the parcels he has to deliver are, on average, heavier. He assumes that the standard deviation is unchanged. He randomly selects 20 parcels and finds that their total mass is 46 kg . Test Jake's belief at the \(5 \%\) level of significance. Jake delivers each parcel to one of three areas, \(A , B\) or \(C\). The probabilities that a parcel has destination area \(A , B\) and \(C\) are \(\frac { 1 } { 2 } , \frac { 1 } { 6 }\) and \(\frac { 1 } { 3 }\) respectively. All parcels are considered to be independent.
5. On a particular day, Jake has three parcels to deliver. Find the probability that he will have to deliver to all three areas.
   
6. On a different day, Jake has two parcels to deliver. Find the probability that he will have to deliver to more than one area.
   

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.

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.

9 A bag contains 100 black discs and 200 white discs. Paula takes five discs at random, without replacement. She notes the number \(X\) of these discs that are black.

1. Find \(\mathrm { P } ( X = 3 )\). Paula decides to use the binomial distribution as a model for the distribution of \(X\).
2. Explain why this model will give probabilities that are approximately, but not exactly, correct.
3. Paula uses the binomial model to find an approximate value for \(\mathrm { P } ( X = 3 )\). Calculate the percentage by which her answer will differ from the answer in part (ii). Paula now assumes that the binomial distribution is a good model for \(X\). She uses a computer simulation to generate 1000 values of \(X\). The number of times that \(X = 3\) occurs is denoted by \(Y\).
4. Calculate estimates of the limits between which two thirds of the values of \(Y\) will lie.

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 \(n\).

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.

9 The probability distribution of a random variable \(X\) is given in the table. Two values of \(X\) are chosen at random. Find the probability that the second value is greater than the first.

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.

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}
   

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 \(\mathrm { P } ( A \mid [ B \cup C ] )\)

The Venn diagram shows the events \(A\), \(B\) and \(C\) and their associated probabilities, where \(p\) and \(q\) are probabilities. \includegraphics[max width=\textwidth, alt={}, center]{29ac0c0b-f963-40a1-beba-7146bbb2d021-02_579_1054_347_447}

1. Find \(\mathrm { P } ( B )\)
2. Determine whether or not \(A\) and \(B\) are independent. Given that \(\mathrm { P } ( C \mid B ) = \mathrm { P } ( C )\)
3. find the value of \(p\) and the value of \(q\) The event \(D\) is such that
   • \(\quad A\) and \(D\) are mutually exclusive
   • \(\mathrm { P } ( B \cap D ) > 0\)
   • On the Venn diagram show a possible position for the event \(D\)

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. 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}
2. 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.
      
   (c) 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)

5 Dafydd, Eli and Fabio are members of an amateur cycling club that holds a time trial each Sunday during the summer. The independent probabilities that Dafydd, Eli and Fabio take part in any one of these trials are \(0.6,0.7\) and 0.8 respectively. Find the probability that, on a particular Sunday during the summer:

1. none of the three cyclists takes part;
2. Fabio is the only one of the three cyclists to take part;
3. exactly one of the three cyclists takes part;
4. either one or two of the three cyclists take part.

4 Each school-day morning, three students, Rita, Said and Ting, travel independently from their homes to the same school by one of three methods: walk, cycle or bus. The table shows the probabilities of their independent daily choices.

1. Calculate the probability that, on any given school-day morning:
   1. all 3 students walk to school;
   2. only Rita travels by bus to school;
   3. at least 2 of the 3 students cycle to school.
2. Ursula, a friend of Rita, never travels to school by bus. The probability that: Ursula walks to school when Rita walks to school is 0.9 ; Ursula cycles to school when Rita cycles to school is 0.7 . Calculate the probability that, on any given school-day morning, Rita and Ursula travel to school by:
   1. the same method;
   2. different methods.

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.