2.03b Probability diagrams: tree, Venn, sample space

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. The following shows the results of a survey on the types of exercise taken by a group of 100 people.

65 run
48 swim
60 cycle
40 run and swim
30 swim and cycle
35 run and cycle
25 do all three

1. Draw a Venn Diagram to represent these data. Find the probability that a randomly selected person from the survey
2. takes none of these types of exercise,
3. swims but does not run,
4. takes at least two of these types of exercise. Jason is one of the above group.
   Given that Jason runs,
5. find the probability that he swims but does not cycle.
   

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 \(\mathrm { P } \left( [ B \cup C ] ^ { \prime } \right)\)

5. A market researcher asked 100 adults which of the three newspapers \(A , B , C\) they read. The results showed that \(30 \operatorname { read } A , 26\) read \(B , 21\) read \(C , 5\) read both \(A\) and \(B , 7\) read both \(B\) and \(C , 6\) read both \(C\) and \(A\) and 2 read all three.

1. Draw a Venn diagram to represent these data. One of the adults is then selected at random.
   Find the probability that she reads
2. at least one of the newspapers,
3. only \(A\),
4. only one of the newspapers,
5. \(A\) given that she reads only one newspaper.

3. For the events \(A\) and \(B\),

1. explain in words the meaning of the term \(\mathrm { P } \left( \begin{array} { l l } B & A \end{array} \right)\),
2. sketch a Venn diagram to illustrate the relationship \(\mathrm { P } \left( \begin{array} { l l } B & A \end{array} \right) = 0\). Three companies operate a bus service along a busy main road. Amber buses run \(50 \%\) of the service and \(2 \%\) of their buses are more than 5 minutes late. Blunder buses run \(30 \%\) of the service and \(10 \%\) of their buses are more than 5 minutes late. Clipper buses run the remainder of the service and only \(1 \%\) of their buses run more than 5 minutes late. Jean is waiting for a bus on the main road.
3. Find the probability that the first bus to arrive is an Amber bus that is more than 5 minutes late. Let \(A , B\) and \(C\) denote the events that Jean catches an Amber bus, a Blunder bus and a Clipper bus respectively. Let \(L\) denote the event that Jean catches a bus that is more than 5 minutes late.
4. Draw a Venn diagram to represent the events \(A , B , \mathrm { C }\) and \(L\). Calculate the probabilities associated with each region and write them in the appropriate places on the Venn diagram.
5. Find the probability that Jean catches a bus that is more than 5 minutes late.

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.

A survey of the reading habits of some students revealed that, on a regular basis, \(25 \%\) read quality newspapers, 45\% read tabloid newspapers and 40\% do not read newspapers at all.

1. Find the proportion of students who read both quality and tabloid newspapers.
2. In the space on page 13 draw a Venn diagram to represent this information. A student is selected at random. Given that this student reads newspapers on a regular basis,
3. find the probability that this student only reads quality newspapers.

1. A disease is known to be present in \(2 \%\) of a population. A test is developed to help determine whether or not someone has the disease.

Given that a person has the disease, the test is positive with probability 0.95
Given that a person does not have the disease, the test is positive with probability 0.03

1. Draw a tree diagram to represent this information. A person is selected at random from the population and tested for this disease.
2. Find the probability that the test is positive. A doctor randomly selects a person from the population and tests him for the disease. Given that the test is positive,
3. find the probability that he does not have the disease.
4. Comment on the usefulness of this test.

5. A person's blood group is determined by whether or not it contains any of 3 substances \(A , B\) and \(C\). A doctor surveyed 300 patients' blood and produced the table below.

1. Draw a Venn diagram to represent this information.
2. Find the probability that a randomly chosen patient's blood contains substance \(C\). Harry is one of the patients. Given that his blood contains substance \(A\),
3. find the probability that his blood contains all 3 substances. Patients whose blood contains none of these substances are called universal blood donors.
4. Find the probability that a randomly chosen patient is a universal blood donor.

2. On a randomly chosen day the probability that Bill travels to school by car, by bicycle or on foot is \(\frac { 1 } { 2 } , \frac { 1 } { 6 }\) and \(\frac { 1 } { 3 }\) respectively. The probability of being late when using these methods of travel is \(\frac { 1 } { 5 } , \frac { 2 } { 5 }\) and \(\frac { 1 } { 10 }\) respectively.

1. Draw a tree diagram to represent this information.
2. Find the probability that on a randomly chosen day
   1. Bill travels by foot and is late,
   2. Bill is not late.
3. Given that Bill is late, find the probability that he did not travel on foot.

2. An experiment consists of selecting a ball from a bag and spinning a coin. The bag contains 5 red balls and 7 blue balls. A ball is selected at random from the bag, its colour is noted and then the ball is returned to the bag. When a red ball is selected, a biased coin with probability \(\frac { 2 } { 3 }\) of landing heads is spun.
When a blue ball is selected a fair coin is spun.

1. Complete the tree diagram below to show the possible outcomes and associated probabilities. \includegraphics[max width=\textwidth, alt={}, center]{039e6fcf-3222-40cc-95ea-37b8dc4a4ddb-03_787_395_734_548} \section*{Coin}
   \includegraphics[max width=\textwidth, alt={}]{039e6fcf-3222-40cc-95ea-37b8dc4a4ddb-03_1007_488_808_950}
   Shivani selects a ball and spins the appropriate coin.
2. Find the probability that she obtains a head. Given that Tom selected a ball at random and obtained a head when he spun the appropriate coin,
3. find the probability that Tom selected a red ball. Shivani and Tom each repeat this experiment.
4. Find the probability that the colour of the ball Shivani selects is the same as the colour of the ball Tom selects.

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.

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. \(\mathrm { P } \left( A ^ { \prime } \cap B \right)\)

7. In a large company, 78\% of employees are car owners, \(30 \%\) of these car owners are also bike owners,
85\% of those who are not car owners are bike owners.

1. Draw a tree diagram to represent this information. An employee is selected at random.
2. Find the probability that the employee is a car owner or a bike owner but not both. Another employee is selected at random. Given that this employee is a bike owner,
3. find the probability that the employee is a car owner. Two employees are selected at random.
4. Find the probability that only one of them is a bike owner.

1. In a factory, three machines, \(J , K\) and \(L\), are used to make biscuits.

Machine \(J\) makes \(25 \%\) of the biscuits. Machine \(K\) makes \(45 \%\) of the biscuits. The rest of the biscuits are made by machine \(L\).
It is known that \(2 \%\) of the biscuits made by machine \(J\) are broken, \(3 \%\) of the biscuits made by machine \(K\) are broken and 5\% of the biscuits made by machine \(L\) are broken.

1. Draw a tree diagram to illustrate all the possible outcomes and associated probabilities. A biscuit is selected at random.
2. Calculate the probability that the biscuit is made by machine \(J\) and is not broken.
3. Calculate the probability that the biscuit is broken.
4. Given that the biscuit is broken, find the probability that it was not made by machine \(K\).
   

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.