2.03a Mutually exclusive and independent events

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 \(R\).

7 A committee of 7 people is to be chosen at random from 18 volunteers.

1. In how many different ways can the committee be chosen? The 18 volunteers consist of 5 people from Gloucester, 6 from Hereford and 7 from Worcester. The committee is to be chosen randomly. Find the probability that the committee will
2. consist of 2 people from Gloucester, 2 people from Hereford and 3 people from Worcester,
3. include exactly 5 people from Worcester,
4. include at least 2 people from each of the three cities. 1 Jenny and John are each allowed two attempts to pass an examination.
5. Jenny estimates that her chances of success are as follows.
   • The probability that she will pass on her first attempt is \(\frac { 2 } { 3 }\).
   • If she fails on her first attempt, the probability that she will pass on her second attempt is \(\frac { 3 } { 4 }\). Calculate the probability that Jenny will pass.
   • John estimates that his chances of success are as follows.
   • The probability that he will pass on his first attempt is \(\frac { 2 } { 3 }\).
   • Overall, the probability that he will pass is \(\frac { 5 } { 6 }\).
   Calculate the probability that if John fails on his first attempt, he will pass on his second attempt. 2 For each of 50 plants, the height, \(h \mathrm {~cm}\), was measured and the value of ( \(h - 100\) ) was recorded. The mean and standard deviation of \(( h - 100 )\) were found to be 24.5 and 4.8 respectively.
6. Write down the mean and standard deviation of \(h\). The mean and standard deviation of the heights of another 100 plants were found to be 123.0 cm and 5.1 cm respectively.
7. Describe briefly how the heights of the second group of plants compare with the first.
8. Calculate the mean height of all 150 plants. 3 In Mr Kendall's cupboard there are 3 tins of baked beans and 2 tins of pineapple. Unfortunately his daughter has removed all the labels for a school project and so the tins are identical in appearance. Mr Kendall wishes to use both tins of pineapple for a fruit salad. He opens tins at random until he has opened the two tins of pineapples. Let \(X\) be the number of tins that Mr Kendall opens.
9. Show that \(\mathrm { P } ( X = 3 ) = \frac { 1 } { 5 }\).
10. The probability distribution of \(X\) is given in the table below.

    \(x\)	2	3	4	5
    \(\mathrm { P } ( X = x )\)	\(\frac { 1 } { 10 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 3 } { 10 }\)	\(\frac { 2 } { 5 }\)

    Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

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.

7 It is given that any integer can be expressed in the form \(3 m + r\), where \(m\) is an integer and \(r\) is 0,1 or 2 . Use this fact to answer the following.

1. By considering the different values of \(r\), prove that the square of any integer cannot be expressed in the form \(3 n + 2\), where \(n\) is an integer.
2. Three integers are chosen at random from the integers 1 to 99 inclusive. The three integers are not necessarily different. By considering the different values of \(r\), determine the probability that the sum of these three integers is divisible by 3 .

15 Two independent events, \(A\) and \(B\), are such that $$\begin{aligned} \mathrm { P } ( A ) & = 0.2 \\ \mathrm { P } ( A \cup B ) & = 0.8 \end{aligned}$$ 15

1. 1. Find \(\mathrm { P } ( B )\) 15
      1. (ii) Find \(\mathrm { P } ( A \cap B )\) 15
   2. State, with a reason, whether or not the events \(A\) and \(B\) are mutually exclusive.

17 The number of toilets in each of a random sample of 200 properties from a town was recorded. Four types of properties were included: terraced, semi-detached, detached and apartment. The data is summarised in the table below. One of the properties is selected at random. \(A\) is the event 'the property has exactly two toilets'. \(B\) is the event 'the property is detached'.
17

1. 1. Find \(\mathrm { P } ( A )\). 17
      1. (ii) Find \(\mathrm { P } \left( A ^ { \prime } \cap B \right)\). 17
   2. (iii) Find \(\mathrm { P } ( A \cup B )\).
      17
   3. Determine whether events \(A\) and \(B\) are independent.
      Fully justify your answer.
      17
   4. Using the table, write down two events, other than event \(\boldsymbol { A }\) and event \(\boldsymbol { B }\), which are mutually exclusive. Event 1 \(\_\_\_\_\) \section*{Event 2}

1. 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".

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.

4. Alyona, Dawn and Sergei are sometimes late for school. The events \(A , D\) and \(S\) are as follows:
A Alyona is late for school
D Dawn is late for school
S Sergei is late for school The Venn diagram below shows the three events \(A , D\) and \(S\) and the probabilities associated with each region of \(D\). The constants \(p , q\) and \(r\) each represent probabilities associated with the three separate regions outside \(D\). \includegraphics[max width=\textwidth, alt={}, center]{b29b0411-8401-420b-9227-befe25c245d8-06_624_1068_845_479}

1. Write down 2 of the events \(A , D\) and \(S\) that are mutually exclusive. Give a reason for your answer. The probability that Sergei is late for school is 0.2 . The events \(A\) and \(D\) are independent.
2. Find the value of \(r\).
   (4) Dawn and Sergei's teacher believes that when Sergei is late for school, Dawn tends to be late for school.
3. State whether or not \(D\) and \(S\) are independent, giving a reason for your answer.
   (1)
4. Comment on the teacher's belief in the light of your answer to part (c).
   (1)
   (Total for Question 4 is 7 marks) \section*{Pearson Edexcel Level 3} \section*{GCE Mathematics} \section*{Paper 2: Mechanics}

   Specimen paper Time: \(\mathbf { 3 5 }\) minutes	Paper Reference(s)
   	\(\mathbf { 8 M A 0 } / \mathbf { 0 2 }\)
   You must have: Mathematical Formulae and Statistical Tables, calculator

   Candidates may use any calculator permitted by Pearson regulations. Calculators must not have the facility for algebraic manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them. \section*{Instructions}
   \section*{Information}
   \section*{Advice}

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)

2. Jenny can choose one of three options, A, B or C, when playing a game. The profit, in pounds, associated with each outcome and their corresponding probabilities are shown on the decision tree in Figure 1. \begin{figure}[h] \end{figure}

1. Calculate the optimal EMV to determine Jenny's best course of action. You must make your working clear. For a profit of \(\pounds x\), Jenny's utility is given by \(1 - \mathrm { e } ^ { - \frac { x } { 400 } }\)
2. Using expected utility as the criterion for the best course of action, determine what Jenny should do now to maximise her profit. You must make your working clear.

2. Mary and Jeff are archers and one morning they play the following game. They shoot an arrow at a target alternately, starting with Mary. The winner is the first to hit the target. You may assume that, with each shot, Mary has a probability 0.25 of hitting the target and Jeff has a probability \(p\) of hitting the target. Successive shots are independent.

1. Determine the probability that Jeff wins the game
   i) with his first shot,
   ii) with his second shot.
2. Show that the probability that Jeff wins the game is $$\frac { 3 p } { 1 + 3 p }$$
3. Find the range of values of \(p\) for which Mary is more likely to win the game than Jeff.

1. A group of 200 adults were asked whether they read cooking magazines, travel magazines or sport magazines.
   Their replies showed that

• 29 read only cooking magazines
• 33 read only travel magazines
• 42 read only sport magazines
• 17 read cooking magazines and sport magazines but not travel magazines
• 11 read travel magazines and sport magazines but not cooking magazines
• 22 read cooking magazines and travel magazines but not sport magazines
• 32 do not read cooking magazines, travel magazines or sport magazines
  1. Using this information, complete the Venn diagram on page 11

One of these adults was chosen at random.

Find the probability that this adult,

1. reads cooking magazines and travel magazines and sport magazines,
2. does not read cooking magazines. Given that this adult reads travel magazines,

find the probability that this adult also reads sport magazines.

\includegraphics[max width=\textwidth, alt={}]{fe416f2e-bc81-444b-a0ca-f0eae9a8b149-11_851_1086_296_493}

1. A box contains only red counters and black counters.

There are \(n\) red counters and \(n + 1\) black counters.
Two counters are selected at random, one at a time without replacement, from the box.

1. Complete the tree diagram for this information. Give your probabilities in terms of \(n\) where necessary. \includegraphics[max width=\textwidth, alt={}, center]{fe416f2e-bc81-444b-a0ca-f0eae9a8b149-24_940_1180_591_413}
2. Show that the probability that the two counters selected are different colours is $$\frac { n + 1 } { 2 n + 1 }$$ The probability that the two counters selected are different colours is \(\frac { 25 } { 49 }\)
3. Find the total number of counters in the box before any counters were selected. Given that the two counters selected are different colours,
4. find the probability that the 1st counter is black. You must show your working.

2 Events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = \frac { 1 } { 2 } , \mathrm { P } ( A \cup B ) = \frac { 5 } { 6 }\) and \(\mathrm { P } ( B \mid A ) = \frac { 1 } { 4 }\).
Find

1. \(\mathrm { P } ( A \cap B )\),
2. \(\mathrm { P } ( B )\).

2 Jill is collecting picture cards given away in packets of a particular brand of breakfast cereal. There are five different cards in the complete set. Each packet contains one card which is equally likely to be any of the five cards in the set.

1. Find the probability that Jill has a complete set of cards from the first five packets that she buys.
2. At some point Jill needs just one more card to complete the set. Let \(X\) be the random variable that represents the number of additional packets that Jill will need to buy in order to complete the set.
   1. Write down the distribution of \(X\).
   2. State the expected number of additional packets that Jill will need to buy.
   3. Find the probability that Jill will need to buy at least 3 additional packets in order to complete the set.

4 At a sixth form college, the student council has 16 members made up as follows. There are 3 male and 3 female students from Year 12, and 6 male and 4 female students from Year 13. Two members of the council are chosen at random to represent the college at conference. Find the probability that the 2 members chosen are

1. the same sex,
2. the same sex and from the same year,
3. from the same year given that they are the same sex.

15 In order to be accepted on a university course, a student needs to pass three exams.
The probability that the student passes the first exam is \(\frac { 3 } { 4 }\).
For each of the second and third exams, the probability of passing the exam is

• the same as the probability of passing the preceding exam if the student passed the preceding exam,
• half of the probability of passing the preceding exam if the student failed the preceding exam.
  1. Draw a tree diagram to represent the above information.
  2. Find the probability that the student passes all three exams.
  3. Find the probability that the student passes at least two of the exams.
  4. Find the probability that the student passes the third exam given that exactly two of the three exams are passed.

Marco has four boxes labelled \(K\), \(L\), \(M\) and \(N\). He places them in a straight line in the order \(K\), \(L\), \(M\), \(N\) with \(K\) on the left. Marco also has four coloured marbles: one is red, one is green, one is white and one is yellow. He places a single marble in each box, at random. Events \(A\) and \(B\) are defined as follows. \(A\): The white marble is in either box \(L\) or box \(M\). \(B\): The red marble is to the left of both the green marble and the yellow marble. Determine whether or not events \(A\) and \(B\) are independent. [3]

Events \(A\) and \(B\) are such that \(\text{P}(A) = 0.3\), \(\text{P}(B) = 0.8\) and \(\text{P}(A \text{ and } B) = 0.4\). State, giving a reason in each case, whether events \(A\) and \(B\) are

1. independent, [2]
2. mutually exclusive. [2]

Two fair twelve-sided dice with sides marked 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 are thrown, and the numbers on the sides which land face down are noted. Events \(Q\) and \(R\) are defined as follows. \(Q\): the product of the two numbers is 24. \(R\): both of the numbers are greater than 8.

1. Find \(\mathrm{P}(Q)\). [2]
2. Find \(\mathrm{P}(R)\). [2]
3. Are events \(Q\) and \(R\) exclusive? Justify your answer. [2]
4. Are events \(Q\) and \(R\) independent? Justify your answer. [2]