2.03c Conditional probability: using diagrams/tables

4 Each weekday, Marta travels to school by bus. Sometimes she arrives late.

• \(L\) is the event that Marta arrives late.
• \(R\) is the event that it is raining.

You are given that \(\mathrm { P } ( L ) = 0.15 , \mathrm { P } ( R ) = 0.22\) and \(\mathrm { P } ( L \mid R ) = 0.45\).

1. Use this information to show that the events \(L\) and \(R\) are not independent.
2. Find \(\mathrm { P } ( L \cap R )\).
3. Draw a Venn diagram showing the events \(L\) and \(R\), and fill in the probability corresponding to each of the four regions of your diagram.

5 Each weekday Alan drives to work. On his journey, he goes over a level crossing. Sometimes he has to wait at the level crossing for a train to pass.

• \(W\) is the event that Alan has to wait at the level crossing.
• \(L\) is the event that Alan is late for work.

You are given that \(\mathrm { P } ( L \mid W ) = 0.4 , \mathrm { P } ( W ) = 0.07\) and \(\mathrm { P } ( L \cup W ) = 0.08\).

1. Calculate \(\mathrm { P } ( L \cap W )\).
2. Draw a Venn diagram, showing the events \(L\) and \(W\). Fill in the probability corresponding to each of the four regions of your diagram.
3. Determine whether the events \(L\) and \(W\) are independent, explaining your method clearly.

1 A survey is being carried out into the carbon footprint of individual citizens. As part of the survey, 100 citizens are asked whether they have attempted to reduce their carbon footprint by any of the following methods.

• Reducing car use
• Insulating their homes
• Avoiding air travel

The numbers of citizens who have used each of these methods are shown in the Venn diagram. \includegraphics[max width=\textwidth, alt={}, center]{e54eba7c-d862-435a-acdd-27df6ede5fab-1_699_1085_849_569} One of the citizens is selected at random.

1. Find the probability that this citizen
   (A) has avoided air travel,
   (B) has used at least two of the three methods.
2. Given that the citizen has avoided air travel, find the probability that this citizen has reduced car use. Three of the citizens are selected at random.
3. Find the probability that none of them have avoided air travel.

3 One train leaves a station each hour. The train is either on time or late. If the train is on time, the probability that the next train is on time is 0.95 . If the train is late, the probability that the next train is on time is 0.6 . On a particular day, the 0900 train is on time.

1. Illustrate the possible outcomes for the 1000,1100 and 1200 trains on a probability tree diagram.
2. Find the probability that
   (A) all three of these trains are on time,
   (B) just one of these three trains is on time,
   (C) the 1200 train is on time.
3. Given that the 1200 train is on time, find the probability that the 1000 train is also on time.

4 In a survey, a large number of young people are asked about their exercise habits. One of these people is selected at random.

• \(G\) is the event that this person goes to the gym.
• \(R\) is the event that this person goes running.

You are given that \(\mathrm { P } ( G ) = 0.24 , \mathrm { P } ( R ) = 0.13\) and \(\mathrm { P } ( G \cap R ) = 0.06\).

1. Draw a Venn diagram, showing the events \(G\) and \(R\), and fill in the probability corresponding to each of the four regions of your diagram.
2. Determine whether the events \(G\) and \(R\) are independent.
3. Find \(\mathrm { P } ( R \mid G )\).

1 Laura frequently flies to business meetings and often finds that her flights are delayed. A flight may be delayed due to technical problems, weather problems or congestion problems, with probabilities \(0.2,0.15\) and 0.1 respectively. The tree diagram shows this information. \includegraphics[max width=\textwidth, alt={}, center]{10679ff3-494d-4f4e-a38a-0832faa91690-1_605_1650_534_284}

1. Write down the values of the probabilities \(a , b\) and \(c\) shown in the tree diagram. One of Laura's flights is selected at random.
2. Find the probability that Laura's flight is not delayed and hence write down the probability that it is delayed.
3. Find the probability that Laura's flight is delayed due to just one of the three problems.
4. Given that Laura's flight is delayed, find the probability that the delay is due to just one of the three problems.
5. Given that Laura's flight has no technical problems, find the probability that it is delayed.
6. In a particular year, Laura has 110 flights. Find the expected number of flights that are delayed.

2 Each day Anna drives to work.

• \(R\) is the event that it is raining.
• \(L\) is the event that Anna arrives at work late.

You are given that \(\mathrm { P } ( R ) = 0.36 , \mathrm { P } ( L ) = 0.25\) and \(\mathrm { P } ( R \cap L ) = 0.2\).

1. Determine whether the events \(R\) and \(L\) are independent.
2. Draw a Venn diagram showing the events \(R\) and \(L\). Fill in the probability corresponding to each of the four regions of your diagram.
3. Find \(\mathrm { P } ( L \mid R )\). State what this probability represents.

3 In the 2001 census, people living in Wales were asked whether or not they could speak Welsh. A resident of Wales is selected at random.

• \(W\) is the event that this person speaks Welsh.
• \(C\) is the event that this person is a child.

You are given that \(\mathrm { P } ( W ) = 0.20 , \mathrm { P } ( C ) = 0.17\) and \(\mathrm { P } ( W \cap C ) = 0.06\).

1. Determine whether the events \(W\) and \(C\) are independent.
2. Draw a Venn diagram, showing the events \(W\) and \(C\), and fill in the probability corresponding to each region of your diagram.
3. Find \(\mathrm { P } ( W \mid C )\).
4. Given that \(\mathrm { P } \left( W \mid C ^ { \prime } \right) = 0.169\), use this information and your answer to part (iii) to comment very briefly on how the ability to speak Welsh differs between children and adults. [1]

1 In a large town, 79\% of the population were born in England, 20\% in the rest of the UK and the remaining \(1 \%\) overseas. Two people are selected at random. You may use the tree diagram below in answering this question. \includegraphics[max width=\textwidth, alt={}, center]{b56ccabe-0e51-4555-b550-78ba347f69bb-1_944_1118_626_547}

1. Find the probability that
   (A) both of these people were born in the rest of the UK,
   (B) at least one of these people was born in England,
   (C) neither of these people was born overseas.
2. Find the probability that both of these people were born in the rest of the UK given that neither was born overseas.
3. (A) Five people are selected at random. Find the probability that at least one of them was not born in England.
   (B) An interviewer selects \(n\) people at random. The interviewer wishes to ensure that the probability that at least one of them was not born in England is more than \(90 \%\). Find the least possible value of \(n\). You must show working to justify your answer.

2 Steve is going on holiday. The probability that he is delayed on his outward flight is 0.3 . The probability that he is delayed on his return flight is 0.2 , independently of whether or not he is delayed on the outward flight.

1. Find the probability that Steve is delayed on his outward flight but not on his return flight.
2. Find the probability that he is delayed on at least one of the two flights.
3. Given that he is delayed on at least one flight, find the probability that he is delayed on both flights.

4 A local council has introduced a recycling scheme for aluminium, paper and kitchen waste. 50 residents are asked which of these materials they recycle. The numbers of people who recycle each type of material are shown in the Venn diagram. \includegraphics[max width=\textwidth, alt={}, center]{b56ccabe-0e51-4555-b550-78ba347f69bb-3_803_804_520_717} One of the residents is selected at random.

1. Find the probability that this resident recycles
   (A) at least one of the materials,
   (B) exactly one of the materials.
2. Given that the resident recycles aluminium, find the probability that this resident does not recycle paper. Two residents are selected at random.
3. Find the probability that exactly one of them recycles kitchen waste.

1 A screening test for a particular disease is applied to everyone in a large population. The test classifies people into three groups: 'positive', 'doubtful' and 'negative'. Of the population, \(3 \%\) is classified as positive, \(6 \%\) as doubtful and the rest negative. In fact, of the people who test positive, only \(95 \%\) have the disease. Of the people who test doubtful, \(10 \%\) have the disease. Of the people who test negative, \(1 \%\) actually have the disease. People who do not have the disease are described as 'clear'.

1. Copy and complete the tree diagram to show this information. \includegraphics[max width=\textwidth, alt={}, center]{f3d936ba-8f60-4350-a5b3-92200996434c-1_833_1156_851_573}
2. Find the probability that a randomly selected person tests negative and is clear.
3. Find the probability that a randomly selected person has the disease.
4. Find the probability that a randomly selected person tests negative given that the person has the disease.
5. Comment briefly on what your answer to part (iv) indicates about the effectiveness of the screening test. Once the test has been carried out, those people who test doubtful are given a detailed medical examination. If a person has the disease the examination will correctly identify this in \(98 \%\) of cases. If a person is clear, the examination will always correctly identify this.
6. A person is selected at random. Find the probability that this person either tests negative originally or tests doubtful and is then cleared in the detailed medical examination.

2 Each day the probability that Ashwin wears a tie is 0.2 . The probability that he wears a jacket is 0.4 . If he wears a jacket, the probability that he wears a tie is 0.3 .

1. Find the probability that, on a randomly selected day, Ashwin wears a jacket and a tie.
2. Draw a Venn diagram, using one circle for the event 'wears a jacket' and one circle for the event 'wears a tie'. Your diagram should include the probability for each region.
3. Using your Venn diagram, or otherwise, find the probability that, on a randomly selected day, Ashwin
   (A) wears either a jacket or a tie (or both),
   (B) wears no tie or no jacket (or wears neither).

3 Isobel plays football for a local team. Sometimes her parents attend matches to watch her play.

• \(A\) is the event that Isobel's parents watch a match.
• \(\quad B\) is the event that Isobel scores in a match.

You are given that \(\frac { 3 } { 7 }\) and \(\mathrm { P } ( A ) = \frac { 7 } { 10 }\).

1. Calculate \(\mathrm { P } ( A \cap B )\). The probability that Isobel does not score and her parents do not attend is 0.1 .
2. Draw a Venn diagram showing the events \(A\) and \(B\), and mark in the probability corresponding to each of the regions of your diagram.
3. Are events \(A\) and \(B\) independent? Give a reason for your answer.
4. By comparing \(\mathrm { P } ( B \mid A )\) with \(\mathrm { P } ( B )\), explain why Isobel should ask her parents not to attend.

4 It has been estimated that \(90 \%\) of paintings offered for sale at a particular auction house are genuine, and that the other \(10 \%\) are fakes. The auction house has a test to determine whether or not a given painting is genuine. If this test gives a positive result, it suggests that the painting is genuine. A negative result suggests that the painting is a fake. If a painting is genuine, the probability that the test result is positive is 0.95 .
If a painting is a fake, the probability that the test result is positive is 0.2 .

1. Copy and complete the probability tree diagram below, to illustrate the information above. \includegraphics[max width=\textwidth, alt={}, center]{f3d936ba-8f60-4350-a5b3-92200996434c-3_466_667_834_737} Calculate the probabilities of the following events.
2. The test gives a positive result.
3. The test gives a correct result.
4. The painting is genuine, given a positive result.
5. The painting is a fake, given a negative result. A second test is more accurate, but very expensive. The auction house has a policy of only using this second test on those paintings with a negative result on the original test.
6. Using your answers to parts (iv) and (v), explain why the auction house has this policy. The probability that the second test gives a correct result is 0.96 whether the painting is genuine or a fake.
7. Three paintings are independently offered for sale at the auction house. Calculate the probability that all three paintings are genuine, are judged to be fakes in the first test, but are judged to be genuine in the second test.

1 A school athletics team has 10 members. The table shows which competitions each of the members can take part in. An athlete is selected at random. Events \(A , B , C , D\) are defined as follows. \(A\) : the athlete can take part in exactly 2 competitions. \(B\) : the athlete can take part in the 200 m . \(C\) : the athlete can take part in the 110 m hurdles. \(D\) : the athlete can take part in the long jump.

1. Write down the value of \(\mathrm { P } ( A \cap B )\).
2. Write down the value of \(\mathrm { P } ( C \cup D )\).
3. Which two of the four events \(A , B , C , D\) are mutually exclusive?
4. Show that events \(B\) and \(D\) are not independent.

2 Jane buys 5 jam doughnuts, 4 cream doughnuts and 3 plain doughnuts.
On arrival home, each of her three children eats one of the twelve doughnuts. The different kinds of doughnut are indistinguishable by sight and so selection of doughnuts is random. Calculate the probabilities of the following events.

1. All 3 doughnuts eaten contain jam.
2. All 3 doughnuts are of the same kind.
3. The 3 doughnuts are all of a different kind.
4. The 3 doughnuts contain jam, given that they are all of the same kind. On 5 successive Saturdays, Jane buys the same combination of 12 doughnuts and her three children eat one each. Find the probability that all 3 doughnuts eaten contain jam on
5. exactly 2 Saturdays out of the 5 ,
6. at least 1 Saturday out of the 5 .

5 The Venn diagram illustrates the occurrence of two events \(A\) and \(B\). \includegraphics[max width=\textwidth, alt={}, center]{64f25a40-d3bf-4212-b92e-655f980c702b-5_480_771_452_655} You are given that \(\mathrm { P } ( A \cap B ) = 0.3\) and that the probability that neither \(A\) nor \(B\) occurs is 0.1 . You are also given that \(\mathrm { P } ( A ) = 2 \mathrm { P } ( B )\). Find \(\mathrm { P } ( B )\).

1 An amateur weather forecaster describes each day as either sunny, cloudy or wet. He keeps a record each day of his forecast and of the actual weather. His results for one particular year are given in the table, A day is selected at random from that year.

1. Show that the probability that the forecast is correct is \(\frac { 264 } { 365 }\). Find the probability that
2. the forecast is correct, given that the forecast is sunny,
3. the forecast is correct, given that the weather is wet,
4. the weather is cloudy, given that the forecast is correct.

1 For the events \(A\) and \(B , \mathrm { P } ( A ) = 0.3 , \mathrm { P } ( B ) = 0.6\) and \(\mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \right) = c\), where \(c \neq 0\).

1. Find \(\mathrm { P } ( A \cap B )\) in terms of \(c\).
2. Find \(\mathrm { P } ( B \mid A )\) and deduce that \(0.1 \leqslant c \leqslant 0.4\).

3 For the events \(A\) and \(B , \mathrm { P } ( A ) = \mathrm { P } ( B ) = \frac { 3 } { 4 }\) and \(\mathrm { P } \left( A \mid B ^ { \prime } \right) = \frac { 1 } { 2 }\).

1. Find \(\mathrm { P } ( A \cap B )\). For a third event \(C , \mathrm { P } ( C ) = \frac { 1 } { 4 }\) and \(C\) is independent of the event \(A \cap B\).
2. Find \(\mathrm { P } ( A \cap B \cap C )\).
3. Given that \(\mathrm { P } ( C \mid A ) = \lambda\) and \(\mathrm { P } ( B \mid C ) = 3 \lambda\), and that no event occurs outside \(A \cup B \cup C\), find the value of \(\lambda\).

8 Events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = 0.3\) and \(\mathrm { P } ( A \mid B ) = 0.6\).

1. Show that \(\mathrm { P } ( B ) \leqslant 0.5\).
2. Given also that \(\mathrm { P } ( A \cup B ) = x\), find \(\mathrm { P } ( B )\) in terms of \(x\).

2 During an outbreak of a disease, it is known that \(68 \%\) of people do not have the disease. Of people with the disease, \(96 \%\) react positively to a test for diagnosing it, as do \(m \%\) of people who do not have the disease.

1. In the case \(m = 8\), find the probability that a randomly chosen person has the disease, given that the person reacts positively to the test.
2. What value of \(m\) would be required for the answer to part (i) to be 0.95 ?

7. In a large college, \(\frac { 3 } { 5 }\) of the students are male, \(\frac { 3 } { 10 }\) of the students are left handed and \(\frac { 1 } { 5 }\) of the male students are left handed. A student is chosen at random.

1. Given that the student is left handed, find the probability that the student is male.
2. Given that the student is female, find the probability that she is left handed.
3. Find the probability that the randomly chosen student is male and right handed. Two students are chosen at random.
4. Find the probability that one student is left handed and one is right handed.

4. Events \(A\) and \(B\) are shown in the Venn diagram below
where \(x , y , 0.10\) and 0.32 are probabilities. \includegraphics[max width=\textwidth, alt={}, center]{c58f3e88-2dbc-40d6-a966-a5765a7c67ba-08_467_798_408_575}

1. Find an expression in terms of \(x\) for
   1. \(\mathrm { P } ( A )\)
   2. \(\mathrm { P } ( B \mid A )\)
2. Find an expression in terms of \(x\) and \(y\) for \(\mathrm { P } ( A \cup B )\) Given that \(\mathrm { P } ( A ) = 2 \mathrm { P } ( B )\)
3. find the value of \(x\) and the value of \(y\)