2.03a Mutually exclusive and independent events

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OCR H240/02 2020 November Q13
8 marks Standard +0.8
Andy and Bev are playing a game.
  • The game consists of three points.
  • On each point, P(Andy wins) = 0.4 and P(Bev wins) = 0.6.
  • If one player wins two consecutive points, then they win the game, otherwise neither player wins.
  1. Determine the probability of the following events.
    1. Andy wins the game. [2]
    2. Neither player wins the game. [3]
Andy and Bev now decide to play a match which consists of a series of games.
  • In each game, if a player wins the game then they win the match.
  • If neither player wins the game then the players play another game.
  1. Determine the probability that Andy wins the match. [3]
OCR H240/02 2023 June Q8
7 marks Easy -1.2
The stem-and-leaf diagram shows the heights, in centimetres, of 15 plants. $\begin{array}{l|l} 0 & 2
1 & 0
2 & 4
3 & 0\ 2\ 4\ 9
4 & 1\ 2\ 4\ 7\ 9
5 & 3\ 7
6 & 2 \end{array}$ Key: \(2 | 5\) means 25 cm.
  1. Draw a box-and-whisker plot to illustrate the data. [4]
A statistician intends to analyse the data, but wants to ignore any outliers before doing so.
  1. Discuss briefly whether there are any heights in the diagram which the statistician should ignore. [3]
AQA AS Paper 2 2020 June Q18
5 marks Moderate -0.8
  1. Bag A contains 7 blue discs, 4 red discs and 1 yellow disc. Two discs are drawn at random from bag A without replacement. Find the probability that exactly one of the discs is blue. [2 marks]
  2. Bag A contains 7 blue discs, 4 red discs and 1 yellow disc. Bag B contains 3 blue discs and 6 red discs. A disc is drawn at random from Bag A and placed in Bag B. A disc is then drawn at random from Bag B. Find the probability that the disc drawn from Bag B is red. [3 marks]
AQA AS Paper 2 2023 June Q15
5 marks Moderate -0.8
Numbered balls are placed in bowls A, B and C In bowl A there are four balls numbered 1, 2, 3 and 7 In bowl B there are eight balls numbered 0, 0, 2, 3, 5, 6, 8 and 9 In bowl C there are nine balls numbered 0, 1, 1, 2, 3, 3, 3, 6 and 7 This information is shown in the diagram below. \includegraphics{figure_15} A three-digit number is generated using the following method: • a ball is selected at random from each bowl • the first digit of the number is the ball drawn from bowl A • the second digit of the number is the ball drawn from bowl B • the third digit of the number is the ball drawn from bowl C
  1. Find the probability that the number generated is even. [1 mark]
  2. Find the probability that the number generated is 703 [2 marks]
  3. Find the probability that the number generated is divisible by 111 [2 marks]
AQA AS Paper 2 Specimen Q15
2 marks Moderate -0.3
A school took 225 children on a trip to a theme park. After the trip the children had to write about their favourite ride at the park from a choice of three. The table shows the number of children who wrote about each ride.
Ride written about
The DropThe BeanstalkThe GiantTotal
Year 724452392
Year 836172275
Year 920132558
Total807570225
Three children were randomly selected from those who went on the trip. Calculate the probability that one wrote about 'The Drop', one wrote about 'The Beanstalk' and one wrote about The Giant'. [2 marks]
AQA Paper 3 2018 June Q14
6 marks Moderate -0.8
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]
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 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 2023 June Q11
1 marks Easy -1.8
A and B are mutually exclusive events. Which one of the following statements must be correct? Tick (\(\checkmark\)) one box. [1 mark] \(P(A \cup B) = P(A) \times P(B)\) \(P(A \cup B) = P(A) - P(B)\) \(P(A \cap B) = 0\) \(P(A \cap B) = 1\)
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]
OCR PURE Q10
5 marks Moderate -0.3
The probability distribution of a random variable \(X\) is given in the table.
\(x\)0246
P\((X = x)\)\(\frac{3}{8}\)\(\frac{5}{16}\)\(4p\)\(p\)
  1. Find the value of \(p\). [2]
  2. Two values of \(X\) are chosen at random. Find the probability that the product of these values is 0. [3]
OCR PURE Q9
4 marks Easy -1.8
In a survey, 50 people were asked whether they had passed A-level English and whether they had passed A-level Mathematics. The numbers of people in various categories are shown in the Venn diagram. \includegraphics{figure_4}
  1. A person is chosen at random from the 50 people. Find the probability that this person has passed A-level Mathematics. [1]
  2. Two people are chosen at random, without replacement, from those who have passed A-level in at least one of the two subjects. Determine the probability that both of these people have passed A-level Mathematics. [3]
OCR MEI AS Paper 2 2018 June Q4
5 marks Moderate -0.8
The probability distribution of the discrete random variable \(X\) is given in Fig. 4.
\(r\)01234
P\((X = r)\)0.20.150.3\(k\)0.25
Fig. 4
  1. Find the value of \(k\). [2]
\(X_1\) and \(X_2\) are two independent values of \(X\).
  1. Find P\((X_1 + X_2 = 6)\). [3]
WJEC Unit 2 2018 June Q02
7 marks Easy -1.3
The Venn diagram shows the subjects studied by 40 sixth form students. \(F\) represents the set of students who study French, \(M\) represents the set of students who study Mathematics and \(D\) represents the set of students who study Drama. The diagram shows the number of students in each set. \includegraphics{figure_2}
  1. Explain what \(M \cap D'\) means in this context. [1]
  2. One of these students is chosen at random. Find the probability that this student studies
    1. exactly two of these subjects,
    2. Mathematics or French or both. [3]
  3. Determine whether studying Mathematics and studying Drama are statistically independent for these students. [3]
WJEC Unit 2 2024 June Q3
8 marks Moderate -0.3
The following Venn diagram shows the participation of 100 students in three activities, \(A\), \(B\), and \(C\), which represent athletics, baseball and climbing respectively. \includegraphics{figure_3} For these 100 students, participation in athletics and participation in climbing are independent events.
  1. Show that \(x = 10\) and find the value of \(y\). [5]
  2. Two students are selected at random, one after the other without replacement. Find the probability that the first student does athletics and the second student does only climbing. [3]
WJEC Unit 2 Specimen Q1
6 marks Moderate -0.8
The events \(A, B\) are such that \(P(A) = 0.2, P(B) = 0.3\). Determine the value of \(P(A \cup B)\) when
  1. \(A,B\) are mutually exclusive, [2]
  2. \(A,B\) are independent, [3]
  3. \(A \subset B\). [1]
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]
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 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 FM Statistics 2021 September Q2
9 marks Moderate -0.3
\(P(E) = 0.25\), \(P(F) = 0.4\) and \(P(E \cap F) = 0.12\)
  1. Find \(P(E'|F')\) [2 marks]
  2. Explain, showing your working, whether or not \(E\) and \(F\) are statistically independent. Give reasons for your answer. [2 marks]
The event \(G\) has \(P(G) = 0.15\) The events \(E\) and \(G\) are mutually exclusive and the events \(F\) and \(G\) are independent.
  1. Draw a Venn diagram to illustrate the events \(E\), \(F\) and \(G\), giving the probabilities for each region. [3 marks]
  2. Find \(P([F \cup G]')\) [2 marks]
SPS SPS FM Statistics 2021 September Q3
11 marks Moderate -0.8
A group of students were surveyed by a principal and \(\frac{2}{3}\) were found to always hand in assignments on time. When questioned about their assignments \(\frac{3}{5}\) said they always start their assignments on the day they are issued and, of those who always start their assignments on the day they are issued, \(\frac{11}{20}\) hand them in on time.
  1. Draw a tree diagram to represent this information. [3 marks]
  2. Find the probability that a randomly selected student:
    1. always start their assignments on the day they are issued and hand them in on time. [2 marks]
    2. does not always hand in assignments on time and does not start their assignments on the day they are issued. [4 marks]
  3. Determine whether or not always starting assignments on the day they are issued and handing them in on time are statistically independent. Give reasons for your answer. [2 marks]
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 2024 January Q7
11 marks Standard +0.8
The probability distribution of a random variable \(X\) is modelled as follows. $$\text{P}(X = x) = \begin{cases} \frac{k}{x} & x = 1, 2, 3, 4, \\ 0 & \text{otherwise,} \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac{12}{25}\). [2]
  2. Show in a table the values of \(X\) and their probabilities. [1]
  3. The values of three independent observations of \(X\) are denoted by \(X_1\), \(X_2\) and \(X_3\). Find P\((X_1 > X_2 + X_3)\). [3]
In a game, a player notes the values of successive independent observations of \(X\) and keeps a running total. The aim of the game is to reach a total of exactly 7.
  1. Determine the probability that a total of exactly 7 is first reached on the 5th observation. [5]
SPS SPS FM Statistics 2025 April Q3
9 marks Standard +0.8
Miguel has six numbered tiles, labelled 2, 2, 3, 3, 4, 4. He selects two tiles at random, without replacement. The variable \(M\) denotes the sum of the numbers on the two tiles.
  1. Show that \(P(M = 6) = \frac{1}{3}\) [2]
The table shows the probability distribution of \(M\)
\(m\)45678
\(P(M = m)\)\(\frac{1}{15}\)\(\frac{4}{15}\)\(\frac{1}{3}\)\(\frac{4}{15}\)\(\frac{1}{15}\)
Miguel returns the two tiles to the collection. Now Sofia selects two tiles at random from the six tiles, without replacement. The variable \(S\) denotes the sum of the numbers on the two tiles that Sofia selects.
  1. Find \(P(M = S)\) [3]
  2. Find \(P(S = 7 | M = S)\) [4]