2.03c Conditional probability: using diagrams/tables

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

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.

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

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.

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.

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) \(\mathrm { P } ( R )\).

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.

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.

The time, in minutes, taken to fly from London to Malaga has a normal distribution with mean 150 minutes and standard deviation 10 minutes.

1. Find the probability that the next flight from London to Malaga takes less than 145 minutes. The time taken to fly from London to Berlin has a normal distribution with mean 100 minutes and standard deviation \(d\) minutes. Given that \(15 \%\) of the flights from London to Berlin take longer than 115 minutes,
2. find the value of the standard deviation \(d\). The time, \(X\) minutes, taken to fly from London to another city has a normal distribution with mean \(\mu\) minutes. Given that \(\mathrm { P } ( X < \mu - 15 ) = 0.35\)
3. find \(\mathrm { P } ( X > \mu + 15 \mid X > \mu - 15 )\).

6. \begin{figure}[h] \end{figure} The Venn diagram in Figure 1 shows three events \(A , B\) and \(C\) and the probabilities associated with each region of \(B\). The constants \(p , q\) and \(r\) each represent probabilities associated with the three separate regions outside \(B\). The events \(A\) and \(B\) are independent.

1. Find the value of \(p\). Given that \(\mathrm { P } ( B \mid C ) = \frac { 5 } { 11 }\)
2. find the value of \(q\) and the value of \(r\).
3. Find \(\mathrm { P } ( A \cup C \mid B )\).

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 \(K\).
   

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.
   

The random variable \(Z \sim \mathrm {~N} ( 0,1 )\) \(A\) is the event \(Z > 1.1\) \(B\) is the event \(Z > - 1.9\) \(C\) is the event \(- 1.5 < Z < 1.5\)

1. Find
   1. \(\mathrm { P } ( A )\)
   2. \(\mathrm { P } ( B )\)
   3. \(\mathrm { P } ( C )\)
   4. \(\mathrm { P } ( A \cup C )\) The random variable \(X\) has a normal distribution with mean 21 and standard deviation 5
2. Find the value of \(w\) such that \(\mathrm { P } ( X > w \mid X > 28 ) = 0.625\)

6. The time, in minutes, taken by men to run a marathon is modelled by a normal distribution with mean 240 minutes and standard deviation 40 minutes.

1. Find the proportion of men that take longer than 300 minutes to run a marathon.
   (3) Nathaniel is preparing to run a marathon. He aims to finish in the first 20\% of male runners.
2. Using the above model estimate the longest time that Nathaniel can take to run the marathon and achieve his aim.
   (3) The time, \(W\) minutes, taken by women to run a marathon is modelled by a normal distribution with mean \(\mu\) minutes. Given that \(\mathrm { P } ( W < \mu + 30 ) = 0.82\)
3. find \(\mathrm { P } ( W < \mu - 30 \mid W < \mu )\)