Basic two-way table probability

1 At a college, the students choose exactly one of tennis, hockey or netball to play. The table shows the numbers of students in Year 1 and Year 2 at the college playing each of these sports. One student is chosen at random from the 120 students. Events \(X\) and \(N\) are defined as follows:
\(X\) : the student is in Year 1
\(N\) : the student plays netball.

1. Find \(\mathrm { P } ( X \mid N )\).
2. Find \(\mathrm { P } ( N \mid X )\).
3. Determine whether or not \(X\) and \(N\) are independent events.
   One of the students who plays netball takes 8 shots at goal. On each shot, the probability that she will succeed is 0.15 , independently of all other shots.
4. Find the probability that she succeeds on fewer than 3 of these shots.

5 Data about employment for males and females in a small rural area are shown in the table. A person from this area is chosen at random. Let \(M\) be the event that the person is male and let \(E\) be the event that the person is employed.

1. Find \(\mathrm { P } ( M )\).
2. Find \(\mathrm { P } ( M\) and \(E )\).
3. Are \(M\) and \(E\) independent events? Justify your answer.
4. Given that the person chosen is unemployed, find the probability that the person is female.

3 Ronnie obtained data about the gross domestic product (GDP) and the birth rate for 170 countries. He classified each GDP and each birth rate as either 'low', 'medium' or 'high'. The table shows the number of countries in each category. One of these countries is chosen at random.

1. Find the probability that the country chosen has a medium GDP.
2. Find the probability that the country chosen has a low birth rate, given that it does not have a medium GDP.
3. State with a reason whether or not the events 'the country chosen has a high GDP' and 'the country chosen has a high birth rate' are exclusive. One country is chosen at random from those countries which have a medium GDP and then a different country is chosen at random from those which have a medium birth rate.
4. Find the probability that both countries chosen have a medium GDP and a medium birth rate.

6 An amateur weather forecaster describes each day as either sunny, cloudy or wet. He keeps a record each day of his forecast and of the actual weather. His results for one particular year are given in the table. A day is selected at random from that year.

1. Show that the probability that the forecast is correct is \(\frac { 264 } { 365 }\). Find the probability that
2. the forecast is correct, given that the forecast is sunny,
3. the forecast is correct, given that the weather is wet,
4. the weather is cloudy, given that the forecast is correct.

1 An amateur weather forecaster describes each day as either sunny, cloudy or wet. He keeps a record each day of his forecast and of the actual weather. His results for one particular year are given in the table, A day is selected at random from that year.

1. Show that the probability that the forecast is correct is \(\frac { 264 } { 365 }\). Find the probability that
2. the forecast is correct, given that the forecast is sunny,
3. the forecast is correct, given that the weather is wet,
4. the weather is cloudy, given that the forecast is correct.

1. A company employs 90 administrators. The length of time that they have been employed by the company and their gender are summarised in the table below.

One of the 90 administrators is selected at random.

1. Find the probability that the administrator is female.
2. Given that the administrator has been employed by the company for less than 4 years, find the probability that this administrator is male.
3. Given that the administrator has been employed by the company for less than 10 years, find the probability that this administrator is male.
4. State, with a reason, whether or not the event 'selecting a male' is independent of the event 'selecting an administrator who has been employed by the company for less than 4 years'.

7 The table shows the numbers of male and female members of a vintage car club who own either a Jaguar or a Bentley. No member owns both makes of car. One member is chosen at random from these 60 members.

1. Given that this member is male, find the probability that he owns a Jaguar. Now two members are chosen at random from the 60 members. They are chosen one at a time, without replacement.
2. Given that the first one of these members is female, find the probability that both own Jaguars.

1. A company has 1825 employees.

The employees are classified as professional, skilled or elementary.
The following table shows

• the number of employees in each classification
• the two areas, \(A\) or \(B\), where the employees live

An employee is chosen at random.
Find the probability that this employee

1. is skilled,
2. lives in area \(B\) and is not a professional. Some classifications of employees are more likely to work from home.
   • \(65 \%\) of professional employees in both area \(A\) and area \(B\) work from home
   • \(40 \%\) of skilled employees in both area \(A\) and area \(B\) work from home
   • \(5 \%\) of elementary employees in both area \(A\) and area \(B\) work from home
   • Event \(F\) is that the employee is a professional
   • Event \(H\) is that the employee works from home
   • Event \(R\) is that the employee is from area \(A\)
   • Using this information, complete the Venn diagram on the opposite page.
   • Find \(\mathrm { P } \left( R ^ { \prime } \cap F \right)\)
   • Find \(\mathrm { P } \left( [ H \cup R ] ^ { \prime } \right)\)
   • Find \(\mathrm { P } ( F \mid H )\)
   \includegraphics[max width=\textwidth, alt={}]{3a09f809-fa28-4b3d-bb69-ea074433bd8f-13_872_1020_294_525}
   Turn over for a spare diagram if you need to redraw your Venn diagram. Only use this diagram if you need to redraw your Venn diagram.
   \includegraphics[max width=\textwidth, alt={}, center]{3a09f809-fa28-4b3d-bb69-ea074433bd8f-15_872_1017_392_525}

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. An employee is selected at random and found to be married.
5. Find the probability that this employee is in production.

3. In a company the 200 employees are classified as full-time workers, part-time workers or contractors.
The table below shows the number of employees in each category and whether they walk to work or use some form of transport. The events \(F , H\) and \(C\) are that an employee is a full-time worker, part-time worker or contractor respectively. Let \(W\) be the event that an employee walks to work. An employee is selected at random.
Find

1. \(\mathrm { P } ( H )\)
2. \(\mathrm { P } \left( [ F \cap W ] ^ { \prime } \right)\)
3. \(\mathrm { P } ( W \mid C )\) Let \(B\) be the event that an employee uses the bus.
   Given that \(10 \%\) of full-time workers use the bus, \(30 \%\) of part-time workers use the bus and \(20 \%\) of contractors use the bus,
4. draw a Venn diagram to represent the events \(F , H , C\) and \(B\),
5. find the probability that a randomly selected employee uses the bus to travel to work.

2 The number of MPs in the House of Commons was 645 at the beginning of August 2009. The genders of these MPs and the political parties to which they belonged are shown in the table.

1. One MP was selected at random for an interview. Calculate, to three decimal places, the probability that the MP was:
   1. a male Conservative;
   2. a male;
   3. a Liberal Democrat;
   4. Labour, given that the MP was female;
   5. male, given that the MP was not Labour.
2. Two female MPs were selected at random for an enquiry. Calculate, to three decimal places, the probability that both MPs were Labour.
3. Three MPs were selected at random for a committee. Calculate, to three decimal places, the probability that exactly one MP was Labour and exactly one MP was Conservative.

5 Roger is an active retired lecturer. Each day after breakfast, he decides whether the weather for that day is going to be fine ( \(F\) ), dull ( \(D\) ) or wet ( \(W\) ). He then decides on only one of four activities for the day: cycling ( \(C\) ), gardening ( \(G\) ), shopping ( \(S\) ) or relaxing \(( R )\). His decisions from day to day may be assumed to be independent. The table shows Roger's probabilities for each combination of weather and activity.

1. Find the probability that, on a particular day, Roger decided:
   1. that it was going to be fine and that he would go cycling;
   2. on either gardening or shopping;
   3. to go cycling, given that he had decided that it was going to be fine;
   4. not to relax, given that he had decided that it was going to be dull;
   5. that it was going to be fine, given that he did not go cycling.
2. Calculate the probability that, on a particular Saturday and Sunday, Roger decided that it was going to be fine and decided on the same activity for both days.

2 The British and Irish Lions 2005 rugby squad contained 50 players. The nationalities and playing positions of these players are shown in the table.

1. A player was selected at random from the squad for a radio interview. Calculate the probability that the player was:
   1. a Welsh back;
   2. English;
   3. not English;
   4. Irish, given that the player was a back;
   5. a forward, given that the player was not Scottish.
2. Four players were selected at random from the squad to visit a school. Calculate the probability that all four players were English.

2 A basket in a stationery store contains a total of 400 marker and highlighter pens. Of the marker pens, some are permanent and the rest are non-permanent. The colours and types of pen are shown in the table. A pen is selected at random from the basket. Calculate the probability that it is:

1. a blue pen;
2. a marker pen;
3. a blue pen or a marker pen;
4. a green pen, given that it is a highlighter pen;
5. a non-permanent marker pen, given that it is a red pen.

1 A large bookcase contains two types of book: hardback and paperback. The number of books of each type in each of four subject categories is shown in the table.

1. A book is selected at random from the bookcase. Calculate the probability that the book is:
   1. a paperback;
   2. not science fiction;
   3. science fiction or a hardback;
   4. a thriller, given that it is a paperback.
2. Three books are selected at random, without replacement, from the bookcase. Calculate, to three decimal places, the probability that one is crime, one is romance and one is science fiction.

3 The table shows the colour of hair and the colour of eyes of a sample of 750 people from a particular population.

3 The table shows, for a random sample of 500 patients attending a dental surgery, the patients' ages, in years, and the NHS charge bands for the patients' courses of treatment. Band 0 denotes the least expensive charge band and band 3 denotes the most expensive charge band.

1. Calculate, to three decimal places, the probability that a patient, selected at random from these 500 patients, was:
   1. aged between 41 and 65;
   2. aged 66 or over and charged at band 2;
   3. aged between 19 and 40 and charged at most at band 1;
   4. aged 41 or over, given that the patient was charged at band 2;
   5. charged at least at band 2, given that the patient was not aged 66 or over.
2. Four patients at this dental surgery, not included in the above 500 patients, are selected at random. Estimate, to three significant figures, the probability that two of these four patients are aged between 41 and 65 and are not charged at band 0 , and the other two patients are aged 66 or over and are charged at either band 1 or band 2.
   [0pt] [5 marks]

5. A College employs 75 teachers, of whom 47 are full-time and the rest are part-time. Of the 39 male teachers at the College, 26 are full-time.

1. Represent this information on a Venn diagram.
2. One teacher is selected at random to be interviewed by an inspector. Find the probability that the teacher chosen
   1. works full-time and is female,
   2. works part-time, given that he is male.
3. Three teachers are selected at random to be observed by an inspector during one day. Find correct to 3 significant figures the probability that
   1. all three teachers chosen work full-time,
   2. at least one of the three teachers chosen is female.

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.

14 A survey was conducted into the health of 120 teachers. The survey recorded whether or not they had suffered from a range of four health issues in the past year. In addition, their physical exercise level was categorised as low, medium or high.
50 teachers had a low exercise level, 40 teachers had a medium exercise level and 30 teachers had a high exercise level. The results of the survey are shown in the table below. 14

1. Find the probability that a randomly selected teacher:
   14
2. 1. suffers from back trouble and has a high exercise level;
      14
3. (ii) suffers from depression.
   14
4. (iii) suffers from stress, given that they have a low exercise level.
   14
5. For teachers in the survey with a low exercise level, explain why the events 'suffers from back trouble' and 'suffers from stress' are not mutually exclusive.