Two-way table probabilities

4 For a group of 250 cars the numbers, classified by colour and country of manufacture, are shown in the table. One car is selected at random from this group. Find the probability that the selected car is

1. a red or silver car manufactured in Korea,
2. not manufactured in Japan.
   \(X\) is the event that the selected car is white. \(Y\) is the event that the selected car is manufactured in Germany.
3. By using appropriate probabilities, determine whether events \(X\) and \(Y\) are independent.

7 In a group of students, the numbers of boys and girls studying Art, Music and Drama are given in the following table. Each of these 160 students is studying exactly one of these subjects.

1. Find the probability that a randomly chosen student is studying Music.
2. Determine whether the events 'a randomly chosen student is a boy' and 'a randomly chosen student is studying Music' are independent, justifying your answer.
3. Find the probability that a randomly chosen student is not studying Drama, given that the student is a girl.
4. Three students are chosen at random. Find the probability that exactly 1 is studying Music and exactly 2 are boys.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1. The table below shows the destination from school of 180 year 11 pupils. Most pupils either continued education, in school or college, or went into some form of employment.

A reporter selects two pupils at random to interview. Given that the first pupil is in school or college, find the probability that both pupils are girls.

2. Jane conducted a survey. She chose a sample of people from three towns, A, B and C. She noted the following information. 400 people were chosen.
230 people were adults.
55 adults were from town A .
65 children were from town A .
35 children were from town B .
150 people were from town \(B\).

1. In the Printed Answer Booklet, complete the two-way frequency table.

   \cline { 2 - 4 } \multicolumn{1}{c|}{}	Town
   \cline { 2 - 4 } \multicolumn{1}{c|}{}	A	B	C	Total
   adult
   child
   Total

2. One of the people is chosen at random.
   1. Find the probability that this person is an adult from town A .
   2. Given that the person is from town A , find the probability that the person is an adult. For another survey, Jane wanted to choose a random sample from the 820 students living in a particular hostel. She numbered the students from 1 to 820 and then generated some random numbers on her calculator. The random numbers were 0.114287562 and 0.081859817 .
      Jane's friend Kareem used these figures to write down the following sample of five student numbers. 114, 142, 428, 287 and 756
      Jane used the same figures to write down the following sample of five student numbers.
      \(114,287,562,81\) and 817
3. 1. State, with a reason, which one of these samples is not random.
   2. Explain why Jane omitted the number 859 from her sample.

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.

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)

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. \(D \cap ( R \cup S )\).

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
2. (ii) Find \(\mathrm { P } \left( A ^ { \prime } \cap B \right)\). 17
3. (iii) Find \(\mathrm { P } ( A \cup B )\).
   17
4. Determine whether events \(A\) and \(B\) are independent.
   Fully justify your answer.
   17
5. 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}