2.03d Calculate conditional probability: from first principles

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 stem and leaf diagram shows the number of deliveries made by Pat each day for 24 days

\begin{table}[h] \end{table} where \(a\), \(b\) and \(c\) are positive integers with \(a < b < c\) An outlier is defined as any value greater than \(1.5 \times\) interquartile range above the upper quartile. Given that there is only one outlier for these data,

1. show that \(c = 9\) The number of deliveries made by Pat each day is represented by \(d\) The data in the stem and leaf diagram are coded using $$x = d - 125$$ and the following summary statistics are obtained $$\sum x = - 96 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 1306$$
2. Find the mean number of deliveries.
3. Find the standard deviation of the number of deliveries. One of these 24 days is selected at random. The random variable \(D\) represents the number of deliveries made by Pat on this day. The random variable \(X = D - 125\)
4. Find \(\mathrm { P } ( D > 118 \mid X < 0 )\)

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

3. Hei and Tang are designing some pieces of art. They collected a large number of sticks. The random variable \(L\) represents the length of a stick in centimetres and has a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). They sorted the sticks into lengths and painted them.
They found that \(60 \%\) of the sticks were longer than 45 cm and these were painted red, whilst \(15 \%\) of the sticks were shorter than 35 cm and these were painted blue. The remaining sticks were painted yellow.

1. Show that \(\mu\) and \(\sigma\) satisfy $$45 + 0.2533 \sigma = \mu$$
2. Find a second equation in \(\mu\) and \(\sigma\).
3. Hence find the value of \(\mu\) and the value of \(\sigma\).
4. Find
   1. \(\mathrm { P } ( L > 35 \mid L < 45 )\)
   2. \(\mathrm { P } ( L < 45 \mid L > 35 )\) Hei created her piece of art using a random selection of blue and yellow sticks.
      Tang created his piece of art using a random selection of red and yellow sticks.
      Hei and Tang each used the same number of sticks to create their piece of art.
      George is viewing Hei's and Tang's pieces of art. He finds a yellow stick on the floor that has fallen from one of these pieces.
5. With reference to your answers to part (d), state, giving a reason, whether the stick is more likely to have fallen from Hei's or Tang's piece of art.

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.

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}

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.

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. 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. 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.