2.03b Probability diagrams: tree, Venn, sample space

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 \(R\).

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}

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}

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.

3. The table below shows the transport options, usual travel times, possible delay times and corresponding probabilities of delay for a journey. All times are in minutes.

1. Draw a decision tree to model the transport options and the possible outcomes.
2. State the minimum expected travel time and the corresponding transport option indicated by the decision tree.

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.

5 In an archery competition, competitors are allowed up to three attempts to hit the bulls-eye. No one who succeeds may try again. \(45 \%\) of those entering the competition hit the bulls-eye first time. For those who fail to hit it the first time, \(60 \%\) of those attempting it for the second time succeed in hitting it. For those who fail twice, only \(15 \%\) of those attempting it for the third time succeed in hitting it. By drawing a tree diagram, or otherwise,

1. find the probability that a randomly chosen competitor fails at all three attempts,
2. find the probability that a randomly chosen competitor fails at the first attempt but succeeds at either the second or third attempt,
3. find the probability that a randomly chosen competitor succeeds in hitting the bulls-eye,
4. find the probability that a randomly chosen competitor requires exactly two attempts given that the competitor is successful.

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.

Nikita goes shopping to buy a birthday present for her mother. She buys either a scarf, with probability 0.3, or a handbag. The probability that her mother will like the choice of scarf is 0.72. The probability that her mother will like the choice of handbag is \(x\). This information is shown on the tree diagram. The probability that Nikita's mother likes the present that Nikita buys is 0.783.

1. Find \(x\). [3]
2. Given that Nikita's mother does not like her present, find the probability that the present is a scarf. [4]

Sharik attempts a multiple choice revision question on-line. There are 3 suggested answers, one of which is correct. When Sharik chooses an answer the computer indicates whether the answer is right or wrong. Sharik first chooses one of the three suggested answers at random. If this answer is wrong he has a second try, choosing an answer at random from the remaining 2. If this answer is also wrong Sharik then chooses the remaining answer, which must be correct.

1. Draw a fully labelled tree diagram to illustrate the various choices that Sharik can make until the computer indicates that he has answered the question correctly. [4]
2. The random variable \(X\) is the number of attempts that Sharik makes up to and including the one that the computer indicates is correct. Draw up the probability distribution table for \(X\) and find E\((X)\). [4]

A bag contains a large number of coloured counters. Each counter is labelled A, B or C 30% of the counters are labelled A 45% of the counters are labelled B The rest of the counters are labelled C It is known that 2% of the counters labelled A are red 4% of the counters labelled B are red 6% of the counters labelled C are red One counter is selected at random from the bag.

1. Complete the tree diagram on the opposite page to illustrate this information. [2]
2. Calculate the probability that the counter is labelled A and is not red. [2]
3. Calculate the probability that the counter is red. [2]
4. Given that the counter is red, find the probability that it is labelled C [3]

\includegraphics{figure_3}

Three events \(A\), \(B\) and \(C\) are such that $$\mathrm{P}(A) = 0.1 \quad \mathrm{P}(B|A) = 0.3 \quad \mathrm{P}(A \cup B) = 0.25 \quad \mathrm{P}(C) = 0.5$$ Given that \(A\) and \(C\) are mutually exclusive

1. find P\((A \cup C)\) [1]
2. Show that P\((B) = 0.18\) [3]

Given also that \(B\) and \(C\) are independent,

1. draw a Venn diagram to represent the events \(A\), \(B\) and \(C\) and the probabilities associated with each region. [5]

A jar contains 2 red, 1 blue and 1 green bead. Two beads are drawn at random from the jar without replacement.

1. In the space below, draw a tree diagram to illustrate all the possible outcomes and associated probabilities. State your probabilities clearly. [3]
2. Find the probability that a blue bead and a green bead are drawn from the jar. [2]

There are 180 students at a college following a general course in computing. Students on this course can choose to take up to three extra options. 112 take systems support, 70 take developing software, 81 take networking, 35 take developing software and systems support, 28 take networking and developing software, 40 take systems support and networking, 4 take all three extra options.

1. In the space below, draw a Venn diagram to represent this information. [5]

A student from the course is chosen at random. Find the probability that this student takes

1. none of the three extra options, [1]
2. networking only. [1]

Students who want to become technicians take systems support and networking. Given that a randomly chosen student wants to become a technician,

1. find the probability that this student takes all three extra options. [2]

Jake and Kamil are sometimes late for school. The events \(J\) and \(K\) are defined as follows \(J =\) the event that Jake is late for school \(K =\) the event that Kamil is late for school \(\text{P}(J) = 0.25\), \(\text{P}(J \cap K) = 0.15\) and \(\text{P}(J' \cap K') = 0.7\) On a randomly selected day, find the probability that

1. at least one of Jake or Kamil are late for school, [1]
2. Kamil is late for school. [2]

Given that Jake is late for school,

1. find the probability that Kamil is late. [3]

The teacher suspects that Jake being late for school and Kamil being late for school are linked in some way.

1. Determine whether or not \(J\) and \(K\) are statistically independent. [2]
2. Comment on the teacher's suspicion in the light of your calculation in (d). [1]

For any married couple who are members of a tennis club, the probability that the husband has a degree is \(\frac{3}{5}\) and the probability that the wife has a degree is \(\frac{1}{2}\). The probability that the husband has a degree, given that the wife has a degree, is \(\frac{11}{12}\). A married couple is chosen at random.

1. Show that the probability that both of them have degrees is \(\frac{11}{24}\). [2]
2. Draw a Venn diagram to represent these data. [5]

Find the probability that

1. only one of them has a degree, [2]
2. neither of them has a degree. [3]

Two married couples are chosen at random.

1. Find the probability that only one of the two husbands and only one of the two wives have degrees. [6]

The values of the two variables \(A\) and \(B\) given in the table in Question 5 are written on cards and placed in two separate packs, which are labelled \(A\) and \(B\). One card is selected from Pack \(A\). Let \(A_i\) represent the event that the first digit on this card is \(i\).

1. Write down the value of P\((A_2)\). [1 mark] The card taken from Pack \(A\) is now transferred into Pack \(B\), and one card is picked at random from Pack \(B\). Let \(B_i\) represent the event that the first digit on this card is \(i\).
2. Show that P\((A_1 \cap B_1) = \frac{1}{24}\). [3 marks]
3. Show that P\((A_6 | B_2) = \frac{4}{41}\). [5 marks]
4. Find the value of P\((A_1 \cup B_4)\). [5 marks]

70% of the households in a town have a CD player and 45% have both a CD player and a personal computer (PC). 18% have neither a CD player nor a PC.

1. Illustrate this information using a Venn diagram. [3 marks]
2. Find the percentage of the households that do not have a PC. [2 marks]
3. Find the probability that a household chosen at random has a CD player or a PC but not both. [2 marks]