2.03d Calculate conditional probability: from first principles

299 questions

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Edexcel S2 Q5
14 marks Standard +0.3
A continuous random variable \(X\) has the cumulative distribution function $$F(x) = 0 \quad x < 2,$$ $$F(x) = k(x - a)^2 \quad 2 \leq x \leq 6,$$ $$F(x) = 1 \quad x \geq 6.$$
  1. Find the values of the constants \(a\) and \(k\). [4 marks]
  2. Show that the median of \(X\) is \(2(1 + \sqrt{2})\). [4 marks]
  3. Given that \(X > 4\), find the probability that \(X > 5\). [6 marks]
AQA S3 2016 June Q2
15 marks Moderate -0.3
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]
OCR H240/02 2020 November Q15
10 marks Challenging +1.2
In this question you must show detailed reasoning. The random variable \(X\) has probability distribution defined as follows. $$P(X = x) = \begin{cases} \frac{15}{64} \times \frac{2^x}{x!} & x = 2, 3, 4, 5, \\ 0 & \text{otherwise}. \end{cases}$$
  1. Show that \(P(X = 2) = \frac{15}{32}\). [1]
The values of three independent observations of \(X\) are denoted by \(X_1\), \(X_2\) and \(X_3\).
  1. Given that \(X_1 + X_2 + X_3 = 9\), determine the probability that at least one of these three values is equal to 2. [6]
Freda chooses values of \(X\) at random until she has obtained \(X = 2\) exactly three times. She then stops.
  1. Determine the probability that she chooses exactly 10 values of \(X\). [3]
OCR H240/02 2023 June Q13
10 marks Easy -1.8
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]
AQA Paper 3 2019 June Q14
7 marks Easy -1.3
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 exerciseMedium exerciseHigh exercise
Back trouble14710
Stress38145
Depression921
Headache/Migraine455
  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]
AQA Paper 3 2020 June Q13
6 marks Easy -1.3
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.
PrimarySecondarySixth-form
Male92423
Female358524
  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]
AQA Paper 3 2021 June Q14
7 marks Standard +0.3
\(A\) 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]
AQA Paper 3 2024 June Q18
7 marks Easy -1.3
The Human Resources director in a company is investigating the graduate status and salaries of its employees. Event \(G\) is defined as the employee is a graduate. Event \(H\) is defined as the employee earns at least £40 000 a year. The director summarised the findings in the table of probabilities below.
\(H\)\(H'\)
\(G\)0.210.18
\(G'\)0.070.54
\begin{enumerate}[label=(\alph*)] \item An employee is selected at random.
  1. Find P(\(G\)) [1 mark]
  2. Find P[\((G \cap H)'\)] [2 marks]
  3. Find P(\(H | G'\)) [2 marks]
\item Determine whether the events \(G\) and \(H\) are independent. Fully justify your answer. [2 marks]
AQA Paper 3 Specimen Q15
6 marks Standard +0.8
A sample of 200 households was obtained from a small town. Each household was asked to complete a questionnaire about their purchases of takeaway food. \(A\) is the event that a household regularly purchases Indian takeaway food. \(B\) is the event that a household regularly purchases Chinese takeaway food. It was observed that \(P(B|A) = 0.25\) and \(P(A|B) = 0.1\) Of these households, 122 indicated that they did not regularly purchase Indian or Chinese takeaway food. A household is selected at random from those in the sample. Find the probability that the household regularly purchases both Indian and Chinese takeaway food. [6 marks]
OCR MEI Paper 2 Specimen Q7
4 marks Moderate -0.8
Two events \(A\) and \(B\) are such that \(\text{P}(A) = 0.6\), \(\text{P}(B) = 0.5\) and \(\text{P}(A \cup B) = 0.85\). Find \(\text{P}(A | B)\). [4]
WJEC Unit 4 2018 June Q1
7 marks Easy -1.2
An architect bids for two construction projects. He estimates the probability of winning bid \(A\) is \(0 \cdot 6\), the probability of winning bid \(B\) is \(0 \cdot 5\) and the probability of winning both is \(0 \cdot 2\).
  1. Show that the probability that he does not win either bid is \(0 \cdot 1\). [2]
  2. Find the probability that he wins exactly one bid. [2]
  3. Given that he does not win bid \(A\), find the probability that he wins bid \(B\). [3]
WJEC Unit 4 2018 June Q2
7 marks Moderate -0.8
  1. Marie is an athlete who competes in the high jump. In a certain competition she is allowed two attempts to clear each height, but if she is successful with the first attempt she does not jump again at this height. The probability that she is successful with her first jump at a height of \(1 \cdot 7\) m is \(p\). The probability that she is successful with her second jump is also \(p\). The probability that she clears \(1 \cdot 7\) m is \(0 \cdot 64\). Find the value of \(p\). [4]
  2. The following table shows the numbers of male and female athletes competing for Wales in track and field events at a competition.
    TrackField
    Male139
    Female74
    Two athletes are chosen at random to participate in a drugs test. Given that the first athlete is male, find the probability that both are field athletes. [3]
WJEC Unit 4 2019 June Q1
5 marks Moderate -0.8
Val buys electrical components from one of 3 suppliers \(A\), \(B\), \(C\), in the ratio \(2:1:7\). The probability that the component is faulty is \(0.33\) for \(A\), \(0.45\) for \(B\) and \(0.05\) for \(C\). Val selects a component at random.
  1. Find the probability that the component works. [3]
  2. Given that the component works, find the probability that Val bought the component from supplier \(B\). [2]
SPS SPS FM Statistics 2021 June Q2
8 marks Moderate -0.3
Events \(A\) and \(B\) are such that \(P(A \cup B) = 0.95\), \(P(A \cap B) = 0.6\) and \(P(A|B) = 0.75\).
  1. Find \(P(B)\). [3]
  2. Find \(P(A)\). [3]
  3. Show that the events \(A'\) and \(B\) are independent. [2]
SPS SPS FM Statistics 2021 June Q6
6 marks Standard +0.3
Only two airlines fly daily into an airport. AMP Air has 70 flights per day and Volt Air has 65 flights per day. Passengers flying with AMP Air have an 18% probability of losing their luggage and passengers flying with Volt Air have a 23% probability of losing their luggage. You overhear a passenger in the airport complaining about her luggage being lost. Find the exact probability that she travelled with Volt Air, giving your answer as a rational number. [6]
SPS SPS SM 2021 February Q3
8 marks Standard +0.3
The Venn diagram shows the probabilities associated with four events, \(A\), \(B\), \(C\) and \(D\) \includegraphics{figure_3}
  1. Write down any pair of mutually exclusive events from \(A\), \(B\), \(C\) and \(D\) [1]
  2. Given that \(P(B) = 0.4\) find the value of \(p\) [1]
  3. Given also that \(A\) and \(B\) are independent find the value of \(q\) [2]
  4. Given further that \(P(B'|C) = 0.64\) find
    1. the value of \(r\)
    2. the value of \(s\)
    [4]
SPS SPS SM Statistics 2024 January Q5
7 marks Standard +0.8
Labrador puppies may be black, yellow or chocolate in colour. Some information about a litter of 9 puppies is given in the table.
malefemale
black13
yellow21
chocolate11
Four puppies are chosen at random to train as guide dogs.
  1. Determine the probability that at least 3 black puppies are chosen. [3]
  2. Determine the probability that exactly 3 females are chosen given that at least 3 black puppies are chosen. [3]
  3. Explain whether the 2 events 'choosing exactly 3 females' and 'choosing at least 3 black puppies' are independent events. [1]
SPS SPS SM Statistics 2025 April Q4
8 marks Moderate -0.3
A manufacturing plant produces electronic circuit boards that need to pass two quality checks - a mechanical inspection and an electrical test. Historical data shows that 15% of boards fail the mechanical inspection. Of those that pass the mechanical inspection, 8% fail the electrical test. Of those that fail the mechanical inspection, 60% fail the electrical test.
  1. If a board is randomly selected from production, what is the probability that it passes both inspections? [2]
  2. If a board is selected at random and is found to have passed the electrical test, what is the probability that it also passed the mechanical inspection? [3]
  3. The company continues to test boards from a large batch until finding one that passes both inspections. Each board is tested independently of all others. What is the probability that they need to test exactly 3 boards to find one that passes both inspections? [3]
SPS SPS SM Statistics 2024 September Q2
4 marks Moderate -0.8
A factory buys 10\% of its components from supplier \(A\), 30\% from supplier \(B\) and the rest from supplier \(C\). It is known that 6\% of the components it buys are faulty. Of the components bought from supplier \(A\), 9\% are faulty and of the components bought from supplier \(B\), 3\% are faulty.
  1. Find the percentage of components bought from supplier \(C\) that are faulty. [3]
A component is selected at random.
  1. Explain why the event "the component was bought from supplier \(B\)" is not statistically independent from the event "the component is faulty". [1]
OCR H240/02 2017 Specimen Q11
8 marks Moderate -0.3
Each of the 30 students in a class plays at least one of squash, hockey and tennis. • 18 students play squash • 19 students play hockey • 17 students play tennis • 8 students play squash and hockey • 9 students play hockey and tennis • 11 students play squash and tennis
  1. Find the number of students who play all three sports. [3]
A student is picked at random from the class.
  1. Given that this student plays squash, find the probability that this student does not play hockey. [1]
Two different students are picked at random from the class, one after the other, without replacement.
  1. Given that the first student plays squash, find the probability that the second student plays hockey. [4]
Pre-U Pre-U 9794/1 2010 June Q12
7 marks Moderate -0.3
  1. Events \(A\) and \(B\) are such that \(\mathrm{P}(A' \cap B') = \frac{1}{6}\).
    1. Find \(\mathrm{P}(A \cup B)\). [2]
    2. Given that \(\mathrm{P}(A | B) = \frac{1}{4}\) and \(\mathrm{P}(B) = \frac{1}{3}\), find \(\mathrm{P}(A \cap B)\) and \(\mathrm{P}(A)\). [3]
  2. In playing the UK Lottery, a set of 6 different integers is chosen irrespective of order from the integers 1 to 49 inclusive. How many different sets of 6 integers can be chosen? [2]
Pre-U Pre-U 9794/3 2013 November Q3
5 marks Moderate -0.8
In a large examination room each candidate has just one electronic calculator.
  • \(G\) is the event that a randomly chosen candidate has a graphical calculator.
  • \(T\) is the event that a randomly chosen candidate has a 'Texio' brand calculator.
You are given the following probabilities. $$\text{P}(G) = 0.65 \quad \text{P}(T) = 0.4 \quad \text{P}(G \cap T) = 0.25$$
  1. Are the events \(G\) and \(T\) independent? Justify your answer with an appropriate calculation. [2]
  2. Find P(\(T | G\)) and explain, in the context of this question, what this probability represents. [3]
Pre-U Pre-U 9794/3 2016 June Q6
5 marks Moderate -0.8
\(A\) and \(B\) are independent events. \(P(A) = \frac{3}{4}\) and \(P(A \cap B) = \frac{1}{4}\). Find \(P(A' \cap B)\) and \(P(A' \cap B')\). [5]
Pre-U Pre-U 9795/2 Specimen Q9
10 marks Standard +0.3
A certain type of fossil occurs at a mean rate of \(0.5\) per square metre at a particular location.
  1. State an assumption that must be made so that the above situation can be modelled by a Poisson distribution. [1]
  2. Find the probability of at least 7 of these fossils occurring in an area of \(10 \text{ m}^2\). [2]
  3. Given that at least 4 such fossils have occurred in an area of \(5 \text{ m}^2\), find the probability that there will be more than 6 found in this area of \(5 \text{ m}^2\). [3]
  4. Find the least area that must be searched in order that the probability of finding at least one fossil of this type is greater than \(0.999\). Give your answer to the nearest square metre. [4]