2.03a Mutually exclusive and independent events

2. A car dealer offers purchasers a three year warranty on a new car. He sells two models, the Zippy and the Nifty. For the first 50 cars sold of each model the number of claims under the warranty is shown in the table below. One of the purchasers is chosen at random. Let \(A\) be the event that no claim is made by the purchaser under the warranty and \(B\) the event that the car purchased is a Nifty.

1. Find \(\mathrm { P } ( A \cap B )\).
2. Find \(\mathrm { P } \left( A ^ { \prime } \right)\). Given that the purchaser chosen does not make a claim under the warranty,
3. find the probability that the car purchased is a Zippy.
4. Show that making a claim is not independent of the make of the car purchased. Comment on this result.

A company assembles drills using components from two sources. Goodbuy supplies \(85 \%\) of the components and Amart supplies the rest. It is known that \(3 \%\) of the components supplied by Goodbuy are faulty and \(6 \%\) of those supplied by Amart are faulty.

1. Represent this information on a tree diagram. An assembled drill is selected at random.
2. Find the probability that it is not faulty.
   

5. Articles made on a lathe are subject to three kinds of defect, \(A , B\) or \(C\). A sample of 1000 articles was inspected and the following results were obtained. \begin{displayquote} 31 had a type \(A\) defect
37 had a type \(B\) defect
42 had a type \(C\) defect
11 had both type \(A\) and type \(B\) defects
13 had both type \(B\) and type \(C\) defects
10 had both type \(A\) and type \(C\) defects
6 had all three types of defect.

1. Draw a Venn diagram to represent these data. \end{displayquote} Find the probability that a randomly selected article from this sample had
2. no defects,
3. no more than one of these defects. An article selected at random from this sample had only one defect.
4. Find the probability that it was a type \(B\) defect. Two different articles were selected at random from this sample.
5. Find the probability that both had type \(B\) defects.

4. A bag contains 9 blue balls and 3 red balls. A ball is selected at random from the bag and its colour is recorded. The ball is not replaced. A second ball is selected at random and its colour is recorded.

1. Draw a tree diagram to represent the information. Find the probability that
   1. the second ball selected is red,
   2. both balls selected are red, given that the second ball selected is red.

6. For the events \(A\) and \(B\), $$\mathrm { P } \left( A \cap B ^ { \prime } \right) = 0.32 , \mathrm { P } \left( A ^ { \prime } \cap B \right) = 0.11 \text { and } \mathrm { P } ( A \cup B ) = 0.65$$

1. Draw a Venn diagram to illustrate the complete sample space for the events \(A\) and \(B\).
2. Write down the value of \(\mathrm { P } ( A )\) and the value of \(\mathrm { P } ( B )\).
3. Find \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\).
4. Determine whether or not \(A\) and \(B\) are independent.

7. The heights of a group of athletes are modelled by a normal distribution with mean 180 cm and a standard deviation 5.2 cm . The weights of this group of athletes are modelled by a normal distribution with mean 85 kg and standard deviation 7.1 kg . Find the probability that a randomly chosen athlete

1. is taller than 188 cm ,
2. weighs less than 97 kg .
   (2)
3. Assuming that for these athletes height and weight are independent, find the probability that a randomly chosen athlete is taller than 188 cm and weighs more than 97 kg .
4. Comment on the assumption that height and weight are independent.

In a factory, machines \(A , B\) and \(C\) are all producing metal rods of the same length. Machine \(A\) produces \(35 \%\) of the rods, machine \(B\) produces \(25 \%\) and the rest are produced by machine \(C\). Of their production of rods, machines \(A , B\) and \(C\) produce \(3 \% , 6 \%\) and \(5 \%\) defective rods respectively.

1. Draw a tree diagram to represent this information.
2. Find the probability that a randomly selected rod is
   1. produced by machine \(A\) and is defective,
   2. is defective.
3. Given that a randomly selected rod is defective, find the probability that it was produced by machine \(C\).

5. The following shows the results of a wine tasting survey of 100 people. \begin{displayquote} 96 like wine \(A\),
93 like wine \(B\),
96 like wine \(C\),
92 like \(A\) and \(B\),
91 like \(B\) and \(C\),
93 like \(A\) and \(C\),
90 like all three wines.

1. Draw a Venn Diagram to represent these data. \end{displayquote} Find the probability that a randomly selected person from the survey likes
2. none of the three wines,
3. wine \(A\) but not wine \(B\),
4. any wine in the survey except wine \(C\),
5. exactly two of the three kinds of wine. Given that a person from the survey likes wine \(A\),
6. find the probability that the person likes wine \(C\).

2. A group of office workers were questioned for a health magazine and \(\frac { 2 } { 5 }\) were found to take regular exercise. When questioned about their eating habits \(\frac { 2 } { 3 }\) said they always eat breakfast and, of those who always eat breakfast \(\frac { 9 } { 25 }\) also took regular exercise. Find the probability that a randomly selected member of the group

1. always eats breakfast and takes regular exercise,
2. does not always eat breakfast and does not take regular exercise.
3. Determine, giving your reason, whether or not always eating breakfast and taking regular exercise are statistically independent.

1. The bag \(P\) contains 6 balls of which 3 are red and 3 are yellow.

The bag \(Q\) contains 7 balls of which 4 are red and 3 are yellow.
A ball is drawn at random from bag \(P\) and placed in bag \(Q\). A second ball is drawn at random from bag \(P\) and placed in bag \(Q\).
A third ball is then drawn at random from the 9 balls in bag \(Q\). The event \(A\) occurs when the 2 balls drawn from bag \(P\) are of the same colour. The event \(B\) occurs when the ball drawn from bag \(Q\) is red.

1. Complete the tree diagram shown below.
   (4) \includegraphics[max width=\textwidth, alt={}, center]{c78ec7b6-dd06-4de1-94c2-052a5577dd10-12_1201_1390_753_269}
2. Find \(\mathrm { P } ( A )\)
3. Show that \(\mathrm { P } ( B ) = \frac { 5 } { 9 }\)
4. Show that \(\mathrm { P } ( A \cap B ) = \frac { 2 } { 9 }\)
5. Hence find \(\mathrm { P } ( A \cup B )\)
6. Given that all three balls drawn are the same colour, find the probability that they are all red.
   (3)

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. \(\mathrm { P } ( A \cup B )\)
2. \(\mathrm { P } \left( A ^ { \prime } \mid B ^ { \prime } \right)\) The event \(C\) has \(\mathrm { P } ( C ) = 0.20\) The events \(A\) and \(C\) are mutually exclusive and the events \(B\) and \(C\) are independent.
3. Find \(\mathrm { P } ( B \cap C )\)
4. Draw a Venn diagram to illustrate the events \(A , B\) and \(C\) and the probabilities for each region.
5. Find \(\mathrm { P } \left( [ B \cup C ] ^ { \prime } \right)\)

1. A fair die has six faces numbered \(1,2,2,3,3\) and 3 . The die is rolled twice and the number showing on the uppermost face is recorded each time.

Find the probability that the sum of the two numbers recorded is at least 5 .
(5)

6. Three events \(A , B\) and \(C\) are defined in the sample space \(S\). The events \(A\) and \(B\) are mutually exclusive and \(A\) and \(C\) are independent.

1. Draw a Venn diagram to illustrate the relationships between the 3 events and the sample space. Given that \(\mathrm { P } ( A ) = 0.2 , \mathrm { P } ( B ) = 0.4\) and \(\mathrm { P } ( A \cup C ) = 0.7\), find
2. \(\mathrm { P } ( A C )\),
3. \(\mathrm { P } ( A \cup B )\),
4. \(\mathrm { P } ( C )\). END

A group of 100 people produced the following information relating to three attributes. The attributes were wearing glasses, being left handed and having dark hair.
Glasses were worn by 36 people, 28 were left handed and 36 had dark hair. There were 17 who wore glasses and were left handed, 19 who wore glasses and had dark hair and 15 who were left handed and had dark hair. Only 10 people wore glasses, were left handed and had dark hair.

1. Represent these data on a Venn diagram. A person was selected at random from this group.
   Find the probability that this person
2. wore glasses but was not left handed and did not have dark hair,
3. did not wear glasses, was not left handed and did not have dark hair,
4. had only two of the attributes,
5. wore glasses given that they were left handed and had dark hair.

7. (a) Given that \(\mathrm { P } ( A ) = a\) and \(\mathrm { P } ( B ) = b\) express \(\mathrm { P } ( A \cup B )\) in terms of \(a\) and \(b\) when

1. \(A\) and \(B\) are mutually exclusive,
2. \(A\) and \(B\) are independent. Two events \(R\) and \(Q\) are such that \(\mathrm { P } \left( R \cap Q ^ { \prime } \right) = 0.15 , \quad \mathrm { P } ( Q ) = 0.35\) and \(\mathrm { P } ( R \mid Q ) = 0.1\) Find the value of
   (b) \(\mathrm { P } ( R \cup Q )\),
   (c) \(\mathrm { P } ( R \cap Q )\),
   (d) \(\mathrm { P } ( R )\).

4. The Venn diagram in Figure 1 shows the number of students in a class who read any of 3 popular magazines \(A , B\) and \(C\). \begin{figure}[h] \end{figure} One of these students is selected at random.

1. Show that the probability that the student reads more than one magazine is \(\frac { 1 } { 6 }\).
2. Find the probability that the student reads \(A\) or \(B\) (or both).
3. Write down the probability that the student reads both \(A\) and \(C\). Given that the student reads at least one of the magazines,
4. find the probability that the student reads \(C\).
5. Determine whether or not reading magazine \(B\) and reading magazine \(C\) are statistically independent.

4. \begin{figure}[h] \end{figure} Figure 1 shows how 25 people travelled to work.
Their travel to work is represented by the events $$\begin{array} { l l } B & \text { bicycle } \\ T & \text { train } \\ W & \text { walk } \end{array}$$

1. Write down 2 of these events that are mutually exclusive. Give a reason for your answer.
2. Determine whether or not \(B\) and \(T\) are independent events. One person is chosen at random.
   Find the probability that this person
3. walks to work,
4. travels to work by bicycle and train.
5. Given that this person travels to work by bicycle, find the probability that they will also take the train.

A manufacturer carried out a survey of the defects in their soft toys. It is found that the probability of a toy having poor stitching is 0.03 and that a toy with poor stitching has a probability of 0.7 of splitting open. A toy without poor stitching has a probability of 0.02 of splitting open.

1. Draw a tree diagram to represent this information.
2. Find the probability that a randomly chosen soft toy has exactly one of the two defects, poor stitching or splitting open.
   (3) The manufacturer also finds that soft toys can become faded with probability 0.05 and that this defect is independent of poor stitching or splitting open. A soft toy is chosen at random.
3. Find the probability that the soft toy has none of these 3 defects.
4. Find the probability that the soft toy has exactly one of these 3 defects.

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.

1. \(\quad A\) and \(B\) are two events such that

$$\mathrm { P } ( B ) = \frac { 1 } { 2 } \quad \mathrm { P } ( A \mid B ) = \frac { 2 } { 5 } \quad \mathrm { P } ( A \cup B ) = \frac { 13 } { 20 }$$

1. Find \(\mathrm { P } ( A \cap B )\).
2. Draw a Venn diagram to show the events \(A , B\) and all the associated probabilities. Find
3. \(\mathrm { P } ( A )\)
4. \(\mathrm { P } ( B \mid A )\)
5. \(\mathrm { P } \left( A ^ { \prime } \cap B \right)\)

8. For the events \(A\) and \(B\), $$\mathrm { P } \left( A ^ { \prime } \cap B \right) = 0.22 \text { and } \mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \right) = 0.18$$

1. Find \(\mathrm { P } ( A )\).
2. Find \(\mathrm { P } ( A \cup B )\). Given that \(\mathrm { P } ( A \mid B ) = 0.6\)
3. find \(\mathrm { P } ( A \cap B )\).
4. Determine whether or not \(A\) and \(B\) are independent.

1. A college has 80 students in Year 12.

20 students study Biology
28 students study Chemistry
30 students study Physics
7 students study both Biology and Chemistry
11 students study both Chemistry and Physics
5 students study both Physics and Biology
3 students study all 3 of these subjects

1. Draw a Venn diagram to represent this information. A Year 12 student at the college is selected at random.
2. Find the probability that the student studies Chemistry but not Biology or Physics.
3. Find the probability that the student studies Chemistry or Physics or both. Given that the student studies Chemistry or Physics or both,
4. find the probability that the student does not study Biology.
5. Determine whether studying Biology and studying Chemistry are statistically independent.
   

4. The Venn diagram shows the probabilities of customer bookings at Harry's hotel. \(R\) is the event that a customer books a room \(B\) is the event that a customer books breakfast \(D\) is the event that a customer books dinner \(u\) and \(t\) are probabilities. \includegraphics[max width=\textwidth, alt={}, center]{e3b92a5b-c0ad-4176-9b05-cb07a44aa265-08_604_1047_696_450}

1. Write down the probability that a customer books breakfast but does not book a room. Given that the events \(B\) and \(D\) are independent
2. find the value of \(t\)
3. hence find the value of \(u\)
4. Find
   1. \(\quad\) P( \(D \mid R \cap B\) )
   2. \(\mathrm { P } \left( D \mid R \cap B ^ { \prime } \right)\) A coach load of 77 customers arrive at Harry's hotel. Of these 77 customers 40 have booked a room and breakfast 37 have booked a room without breakfast
5. Estimate how many of these 77 customers will book dinner.

The Venn diagram shows three events \(A , B\) and \(C\), where \(p , q , r , s\) and \(t\) are probabilities. \includegraphics[max width=\textwidth, alt={}, center]{319667e7-3f8b-4a33-8fc5-ef72154d1421-10_647_972_306_488}
(b) Find the value of \(r\).
(c) Hence write down the value of \(s\) and the value of \(t\).
(d) State, giving a reason, whether or not the events \(A\) and \(B\) are independent.
(e) Find \(\mathrm { P } ( B \mid A \cup C )\). \(\mathrm { P } ( A ) = 0.5 , \mathrm { P } ( B ) = 0.6\) and \(\mathrm { P } ( C ) = 0.25\) and the events \(B\) and \(C\) are independent.
(a) Find the value of \(p\) and the value of \(q\).