2.03d Calculate conditional probability: from first principles

3. A certain disease occurs in a population in 2 mutually exclusive types. It is difficult to diagnose people with type \(A\) of the disease and there is an unknown proportion \(p\) of the population with type \(A\).
It is easier to diagnose people with type \(B\) of the disease and it is known that \(2 \%\) of the population have type \(B\). A test has been developed to help diagnose whether or not a person has the disease. The event \(T\) represents a positive result on the test. After a large-scale trial of the test, the following information was obtained. For a person with type \(B\) of the disease the probability of a positive test result is 0.96 For a person who does not have the disease the probability of a positive test result is 0.05 For a person with type \(A\) of the disease the probability of a positive test result is \(q\)

1. Complete the tree diagram. \includegraphics[max width=\textwidth, alt={}, center]{9ac7647f-b291-4a64-9518-fa6438a0cc7d-08_776_965_1050_484} The probability of a randomly selected person having a positive test result is 0.169 For a person with a positive test result, the probability that they do not have the disease is \(\frac { 41 } { 169 }\)
2. Find the value of \(p\) and the value of \(q\). A doctor is about to see a person who she knows does not have type \(B\) of the disease but does have a positive test result.
3. 1. Find the probability that this person has type \(A\) of the disease.
   2. State, giving a reason, whether or not the doctor will find the test useful.

2. In the Venn diagram below, \(A , B\) and \(C\) are events and \(p , q , r\) and \(s\) are probabilities. The events \(A\) and \(C\) are independent and \(\mathrm { P } ( A ) = 0.65\) \includegraphics[max width=\textwidth, alt={}, center]{a439724e-b570-434d-bf75-de2b50915042-04_373_815_397_568}

1. State which two of the events \(A\), \(B\) and \(C\) are mutually exclusive.
2. Find the value of \(r\) and the value of \(s\). The events ( \(A \cap C ^ { \prime }\) ) and ( \(B \cup C\) ) are also independent.
3. Find the exact value of \(p\) and the exact value of \(q\). Give your answers as fractions.

1. The events \(H\) and \(W\) are such that

$$\mathrm { P } ( H ) = \frac { 3 } { 8 } \quad \mathrm { P } ( H \cup W ) = \frac { 3 } { 4 }$$ Given that \(H\) and \(W\) are independent,

1. show that \(\mathrm { P } ( W ) = \frac { 3 } { 5 }\) The event \(N\) is such that $$\mathrm { P } ( N ) = \frac { 1 } { 15 } \quad \mathrm { P } ( H \cap N ) = \mathrm { P } ( N )$$
2. Find \(\mathrm { P } \left( N ^ { \prime } \mid H \right)\) Given that \(W\) and \(N\) are mutually exclusive,
3. draw a Venn diagram to represent the events \(H , W\) and \(N\) giving the exact probabilities of each region in the Venn diagram.

A manufacturer fills bottles with oil. The volume of oil in a bottle, \(V \mathrm { ml }\), is normally distributed with \(V \sim \mathrm {~N} \left( 100,2.5 ^ { 2 } \right)\)

1. Find \(\mathrm { P } ( V > 104.9 )\)
2. In a pack of 150 bottles, find the expected number of bottles containing more than 104.9 ml
3. Find the value of \(v\), to 2 decimal places, such that \(\mathrm { P } ( V > v \mid V < 104.9 ) = 0.2801\)

1. A competition consists of two rounds.

The time, in minutes, taken by adults to complete round one is modelled by a normal distribution with mean 15 minutes and standard deviation 2 minutes.

1. Use standardisation to find the proportion of adults that take less than 18 minutes to complete round one. Only the fastest \(60 \%\) of adults from round one take part in round two.
2. Use standardisation to find the longest time that an adult can take to complete round one if they are to take part in round two. The time, \(T\) minutes, taken by adults to complete round two is modelled by a normal distribution with mean \(\mu\) Given that \(\mathrm { P } ( \mu - 10 < T < \mu + 10 ) = 0.95\)
3. find \(\mathrm { P } ( T > \mu - 5 \mid T > \mu - 10 )\)

The Venn diagram shows the probabilities related to teenagers playing 3 particular board games. \(C\) is the event that a teenager plays Chess \(S\) is the event that a teenager plays Scrabble \(G\) is the event that a teenager plays Go
where \(p\) and \(q\) are probabilities. \includegraphics[max width=\textwidth, alt={}, center]{ee0c7c12-84f3-479c-b36a-3357f8529a1c-22_684_935_598_566}

1. Find the probability that a randomly selected teenager plays Chess but does not play Go. Given that the events \(C\) and \(S\) are independent,
2. find the value of \(p\)
3. Hence find the value of \(q\)
4. Find (i) \(\mathrm { P } \left( ( C \cup S ) \cap G ^ { \prime } \right)\) (ii) \(\mathrm { P } ( C \mid ( S \cap G ) )\) A youth club consists of a large number of teenagers.
   In this youth club 76 teenagers play Chess and Go.
5. Use the information in the Venn diagram to estimate how many of the teenagers in the youth club do not play Scrabble.

1. Hugo recorded the purchases of 80 customers in the ladies fashion department of a large store. His results were as follows

20 customers bought a coat
12 customers bought a coat and a scarf
23 customers bought a pair of gloves
13 customers bought a pair of gloves and a scarf no customer bought a coat and a pair of gloves 14 customers did not buy a coat nor a scarf nor a pair of gloves.

1. Draw a Venn diagram to represent all of this information.
2. One of the 80 customers is selected at random.
   1. Find the probability that the customer bought a scarf.
   2. Given that the customer bought a coat, find the probability that the customer also bought a scarf.
   3. State, giving a reason, whether or not the event 'the customer bought a coat' and the event 'the customer bought a scarf' are statistically independent. Hugo had asked the member of staff selling coats and the member of staff selling gloves to encourage customers also to buy a scarf.
3. By considering suitable conditional probabilities, determine whether the member of staff selling coats or the member of staff selling gloves has the better performance at selling scarves to their customers. Give a reason for your answer.

4. Pieces of wood cladding are produced by a timber merchant. There are three types of fault, \(A , B\) and \(C\), that can appear in each piece of wood cladding. The Venn diagram shows the probabilities of a piece of wood cladding having the various types of fault. \includegraphics[max width=\textwidth, alt={}, center]{0377c6e9-ab4f-477d-9236-0732fe81f25e-14_602_1120_497_413} A piece of wood cladding is chosen at random.

1. Find the probability that the piece of wood cladding has more than one type of fault. Fault types \(A\) and \(C\) occur independently.
2. Find the probability that the piece of wood cladding has no faults. Given that the piece of wood cladding has fault \(A\),
3. find the probability that it also has fault \(B\) but not fault \(C\). Two pieces of the wood cladding are selected at random.
4. Find the probability that both have exactly 2 types of fault.

The Venn diagram shows the events \(A , B , C\) and \(D\), where \(p , q , r\) and \(s\) are probabilities. \includegraphics[max width=\textwidth, alt={}, center]{1fda59cb-059e-4850-810f-cc3e69bc058e-20_504_826_296_621}

1. Write down the value of
   1. \(\mathrm { P } ( A )\)
   2. \(\mathrm { P } ( A \mid B )\)
   3. \(\mathrm { P } ( A \mid C )\) Given that \(\mathrm { P } \left( B ^ { \prime } \cap D ^ { \prime } \right) = \frac { 7 } { 10 }\) and \(\mathrm { P } ( C \mid D ) = \frac { 3 } { 5 }\)
2. find the exact value of \(q\) and the exact value of \(r\) Given also that \(\mathrm { P } \left( B \cup C ^ { \prime } \right) = \frac { 5 } { 8 }\)
3. find the exact value of \(s\)

1. Sally plays a game in which she can either win or lose.

A turn consists of up to 3 games. On each turn Sally plays the game up to 3 times. If she wins the first 2 games or loses the first 2 games, then she will not play the 3rd game.

• The probability that Sally wins the first game in a turn is 0.7
• If Sally wins a game the probability that she wins the next game is 0.6
• If Sally loses a game the probability that she wins the next game is 0.2
  1. Use this information to complete the tree diagram on page 3
  2. Find the probability that Sally wins the first 2 games in a turn.
  3. Find the probability that Sally wins exactly 2 games in a turn.

Given that Sally wins 2 games in a turn,

find the probability that she won the first 2 games. Given that Sally won the first game in a turn,

find the probability that she won 2 games. 1st game 2nd game win

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]{61983561-79f7-4883-8ae7-ab1f4955d444-04_785_385_744_568} \includegraphics[max width=\textwidth, alt={}, center]{61983561-79f7-4883-8ae7-ab1f4955d444-04_1054_483_760_954} 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 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.

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.

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.

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 \(C\).

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

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.

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 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.