2.03c Conditional probability: using diagrams/tables

Andy can walk to work, travel by bike or travel by bus. The tree diagram shows the probabilities of any day being dry or wet and the corresponding probabilities for each of Andy's methods of travel. \includegraphics{figure_5} A day is selected at random. Find the probability that

1. the weather is wet and Andy travels by bus, [2]
2. Andy walks or travels by bike, [3]
3. the weather is dry given that Andy walks or travels by bike. [3]

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{figure_6} One of the citizens is selected at random.

1. Find the probability that this citizen
   1. has avoided air travel, [1]
   2. has used at least two of the three methods. [2]
2. Given that the citizen has avoided air travel, find the probability that this citizen has reduced car use. [2]

Three of the citizens are selected at random.

1. Find the probability that none of them have avoided air travel. [3]

In a recent survey, a large number of working people were asked whether they worked full-time or part-time, with part-time being defined as less than 25 hours per week. One of the respondents is selected at random. • \(W\) is the event that this person works part-time. • \(F\) is the event that this person is female. You are given that P(\(W\)) = 0.14, P(\(F\)) = 0.41 and P(\(W \cap F\)) = 0.11.

1. Draw a Venn diagram showing the events \(W\) and \(F\), and fill in the probability corresponding to each of the four regions of your diagram. [3]
2. Determine whether the events \(W\) and \(F\) are independent. [2]
3. Find P(\(W\) | \(F\)) and explain what this probability represents. [3]

Candidates applying for jobs in a large company take an aptitude test, as a result of which they are either accepted, rejected or retested, with probabilities 0.2, 0.5 and 0.3 respectively. When a candidate is retested for the first time, the three possible outcomes and their probabilities remain the same as for the original test. When a candidate is retested for the second time there are just two possible outcomes, accepted or rejected, with probabilities 0.4 and 0.6 respectively.

1. Draw a probability tree diagram to illustrate the outcomes. [3]
2. Find the probability that a randomly selected candidate is accepted. [2]
3. Find the probability that a randomly selected candidate is retested at least once, given that this candidate is accepted. [3]

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. [1]
2. Find \(\mathrm{P}(L \cap R)\). [2]
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. [3]

The events \(A\) and \(B\) are such that $$\text{P}(A) = \frac{5}{16}, \text{P}(B) = \frac{1}{2} \text{ and P}(A|B) = \frac{1}{4}$$ Find

1. P\((A \cap B)\). [2]
2. P\((B'|A)\). [3]
3. P\((A' \cup B)\). [2]
4. Determine, with a reason, whether or not the events \(A\) and \(B\) are independent. [3]

The events \(A\) and \(B\) are such that $$\text{P}(A) = \frac{7}{12}, \quad \text{P}(A \cap B) = \frac{1}{4} \quad \text{and} \quad \text{P}(A|B) = \frac{2}{3}.$$ Find

1. P\((B)\), [3 marks]
2. P\((A \cup B)\), [3 marks]
3. P\((B|A')\). [3 marks]

The events \(A\) and \(B\) are independent and such that $$\text{P}(A) = 2\text{P}(B) \text{ and } \text{P}(A \cap B) = \frac{1}{8}.$$

1. Show that \(\text{P}(B) = \frac{1}{4}\). [5 marks]
2. Find \(\text{P}(A \cup B)\). [3 marks]
3. Find \(\text{P}(A | B')\). [2 marks]

In a food survey, a large number of people are asked whether they like tomato soup, mushroom soup, both or neither. One of these people is selected at random. • \(T\) is the event that this person likes tomato soup. • \(M\) is the event that this person likes mushroom soup. You are given that \(\text{P}(T) = 0.55\), \(\text{P}(M) = 0.33\) and \(\text{P}(T|M) = 0.80\).

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

In a recent survey, a large number of working people were asked whether they worked full-time or part-time, with part-time being defined as less than 25 hours per week. One of the respondents is selected at random. • \(W\) is the event that this person works part-time. • \(F\) is the event that this person is female. You are given that \(\text{P}(W) = 0.14\), \(\text{P}(F) = 0.41\) and \(\text{P}(W \cap F) = 0.11\).

1. Draw a Venn diagram showing the events \(W\) and \(F\), and fill in the probability corresponding to each of the four regions of your diagram. [3]
2. Determine whether the events \(W\) and \(F\) are independent. [2]
3. Find \(\text{P}(W|F)\) and explain what this probability represents. [3]

Andy can walk to work, travel by bike or travel by bus. The tree diagram shows the probabilities of any day being dry or wet and the corresponding probabilities for each of Andy's methods of travel. \includegraphics{figure_7} A day is selected at random. Find the probability that

1. the weather is wet and Andy travels by bus, [2]
2. Andy walks or travels by bike, [3]
3. the weather is dry given that Andy walks or travels by bike. [3]

A plane flies regularly between airports D and T with an intermediate stop at airport M. The time of the plane's departure from, or arrival at, each airport is classified as either early, on time, or late. On 90\% of flights, the plane departs from D on time, and on 10\% of flights, it departs from D late. Of those flights that depart from D on time, 65\% then depart from M on time and 35\% depart from M late. Of those flights that depart from D late, 15\% then depart from M on time and 85\% depart from M late. Any flight that departs from M on time has probability 0.25 of arriving at T early, probability 0.60 of arriving at T on time and probability 0.15 of arriving at T late. Any flight that departs from M late has probability 0.10 of arriving at T early, probability 0.20 of arriving at T on time and probability 0.70 of arriving at T late.

1. Represent this information by a tree diagram on which labels and percentages or probabilities are shown. [3 marks]
2. Hence, or otherwise, calculate the probability that the plane:
   1. arrives at T on time;
   2. arrives at T on time, given that it departed from D on time;
   3. does not arrive at T late, given that it departed from D on time;
   4. does not arrive at T late, given that it departed from M on time.
   [8 marks]
3. Three independent flights of the plane depart from D on time. Calculate the probability that two flights arrive at T on time and that one flight arrives at T early. [4 marks]

The scatter diagram uses information about all the Local Authorities (LAs) in the UK, taken from the 2011 census. For each LA it shows the percentage (\(x\)) of employees who used public transport to travel to work and the percentage (\(y\)) who used motorised private transport. "Public transport" includes train, bus, minibus, coach, underground, metro and light rail. "Motorised private transport" includes car, van, motorcycle, scooter, moped, taxi and passenger in a car or van. \includegraphics{figure_13}

1. Most of the points in the diagram lie on or near the line with equation \(x + y = k\), where \(k\) is a constant.
   1. Give a possible value for \(k\). [1]
   2. Hence give an approximate value for the percentage of employees who either worked from home or walked or cycled to work. [1]
2. The average amount of fuel used per person per day for travelling to work in any LA is denoted by F. Consider the two groups of LAs where the percentages using motorised private transport are highest and lowest.
   1. Using only the information in the diagram, suggest, with a reason, which of these two groups will have greater values of F than the other group. [1]
   A student says that it is not possible to give a reliable answer to part (b)(i) without some further information.
   1. Suggest two kinds of further information which would enable a more reliable answer to be given. [2]
3. Points \(A\) and \(B\) in the diagram are the most extreme outliers. Use their positions on the diagram to answer the following questions about the two LAs represented by these two points.
   1. The two LAs share a certain characteristic. Describe, with a justification, this characteristic. [2]
   2. The environments in these two LAs are very different. Describe, with a justification, this difference. [2]
4. A student says that it is difficult to extract detailed information from the scatter diagram. Explain whether you agree with this criticism. [1]

A teacher in a college asks her mathematics students what other subjects they are studying. She finds that, of her 24 students: 12 study physics 8 study geography 4 study geography and physics

1. A student is chosen at random from the class. Determine whether the event 'the student studies physics' and the event 'the student studies geography' are independent. [2 marks]
2. It is known that for the whole college: the probability of a student studying mathematics is \(\frac{1}{5}\) the probability of a student studying biology is \(\frac{1}{6}\) the probability of a student studying biology given that they study mathematics is \(\frac{3}{8}\) Calculate the probability that a student studies mathematics or biology or both. [4 marks]

A survey was conducted into the health of 120 teachers. The survey recorded whether or not they had suffered from a range of four health issues in the past year. In addition, their physical exercise level was categorised as low, medium or high. 50 teachers had a low exercise level, 40 teachers had a medium exercise level and 30 teachers had a high exercise level. The results of the survey are shown in the table below.

	Low exercise	Medium exercise	High exercise
Back trouble	14	7	10
Stress	38	14	5
Depression	9	2	1
Headache/Migraine	4	5	5

1. Find the probability that a randomly selected teacher:
   1. suffers from back trouble and has a high exercise level; [1 mark]
   2. suffers from depression. [2 marks]
   3. suffers from stress, given that they have a low exercise level. [2 marks]
2. For teachers in the survey with a low exercise level, explain why the events 'suffers from back trouble' and 'suffers from stress' are not mutually exclusive. [2 marks]

Diedre is a head teacher in a school which provides primary, secondary and sixth-form education. There are 200 teachers in her school. The number of teachers in each level of education along with their gender is shown in the table below.

	Primary	Secondary	Sixth-form
Male	9	24	23
Female	35	85	24

1. A teacher is selected at random. Find the probability that:
   1. the teacher is female [1 mark]
   2. the teacher is not a sixth-form teacher. [1 mark]
2. Given that a randomly chosen teacher is male, find the probability that this teacher is not a primary teacher. [2 marks]
3. Diedre wants to select three different teachers at random to be part of a school project. Calculate the probability that all three chosen are secondary teachers. [2 marks]

\(A\) and \(B\) are two events such that $$P(A \cap B) = 0.1$$ $$P(A' \cap B') = 0.2$$ $$P(B) = 2P(A)$$

1. Find \(P(A)\) [4 marks]
2. Find \(P(B|A)\) [2 marks]
3. Determine if \(A\) and \(B\) are independent events. [1 mark]

A sample of 240 households were asked which, if any, of the following animals they own as pets: • cats (C) • dogs (D) • tortoises (T) The results are shown in the table below.

1. Represent this information by fully completing the Venn diagram below. [3 marks] \includegraphics{figure_16}
2. A household is chosen at random from the sample.
   1. Find the probability that the household owns a cat only. [1 mark]
   2. Find the probability that the household owns at least two of the three types of pet. [2 marks]
   3. Find the probability that the household owns a cat or a dog or both, given that the household does not own a tortoise. [2 marks]
3. Determine whether a household owning a cat and a household owning a tortoise are independent of each other. Fully justify your answer. [2 marks]

There are two types of coins in a money box: • 20% are bronze coins • 80% are silver coins Craig takes out a coin at random and places it back in the money box. Craig then takes out a second coin at random.

1. Find the probability that both coins were of the same type. [2 marks]
2. Find the probability that both coins are bronze, given that at least one of the coins is bronze. [2 marks]