Principle of Inclusion/Exclusion

5. A market researcher asked 100 adults which of the three newspapers \(A , B , C\) they read. The results showed that \(30 \operatorname { read } A , 26\) read \(B , 21\) read \(C , 5\) read both \(A\) and \(B , 7\) read both \(B\) and \(C , 6\) read both \(C\) and \(A\) and 2 read all three.

1. Draw a Venn diagram to represent these data. One of the adults is then selected at random.
   Find the probability that she reads
2. at least one of the newspapers,
3. only \(A\),
4. only one of the newspapers,
5. \(A\) given that she reads only one newspaper.

6 Whitefly, blight and mosaic virus are three problems which can affect tomato plants. 100 tomato plants are examined for these problems. The numbers of plants with each type of problem are shown in the Venn diagram. 47 of the plants have none of the problems.
\includegraphics[max width=\textwidth, alt={}, center]{e54eba7c-d862-435a-acdd-27df6ede5fab-3_654_804_1262_699}

1. One of the 100 plants is selected at random. Find the probability that this plant has
   (A) at most one of the problems,
   (B) exactly two of the problems.
2. Three of the 100 plants are selected at random. Find the probability that all of them have at least one of the problems.

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 \(\mathrm { P } ( B )\).

2 In a group of 30 teenagers, 13 of the 18 males watch 'Kops are Kids' on television and 3 of the 12 females watch 'Kops are Kids'.

1. Find the probability that a person chosen at random from the group is either female or watches 'Kops are Kids' or both.
2. Showing your working, determine whether the events 'the person chosen is male' and 'the person chosen watches Kops are Kids' are independent or not.

1 For the events \(A\) and \(B\) it is given that $$\mathrm { P } ( A ) = 0.6 , \mathrm { P } ( B ) = 0.3 \text { and } \mathrm { P } ( A \text { or } B \text { but not both } ) = 0.4 \text {. }$$

1. Find \(\mathrm { P } ( A \cap B )\).
2. Find \(\mathrm { P } \left( A ^ { \prime } \cap B \right)\).
3. State, giving a reason, whether \(A\) and \(B\) are independent.

1. The Venn diagram shows three events, \(A\), \(B\) and \(C\), and their associated probabilities.
   \includegraphics[max width=\textwidth, alt={}, center]{e73193ee-339e-48ab-811c-9ab6817f786d-04_680_780_296_644}

Events \(B\) and \(C\) are mutually exclusive.
Events \(A\) and \(C\) are independent.
Showing your working, find the value of \(x\), the value of \(y\) and the value of \(z\).

5. A group of 100 students are asked if they like folk music, rock music or soul music. \begin{displayquote} All students who like folk music also like rock music No students like both rock music and soul music 75 students do not like soul music 12 students who like rock music do not like folk music 30 students like folk music

1. Draw a Venn diagram to illustrate this information.
2. State two of these types of music that are mutually exclusive. \end{displayquote} Find the probability that a randomly chosen student
3. does not like folk music, rock music or soul music,
4. likes rock music,
5. likes folk music or soul music. Given that a randomly chosen student likes rock music,
6. find the probability that he or she also likes folk music.

5. The events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = \frac { 1 } { 2 } , \mathrm { P } ( B ) = \frac { 1 } { 3 }\) and \(\mathrm { P } ( A \cap B ) = \frac { 1 } { 4 }\).

1. Using the space below, represent these probabilities in a Venn diagram. Hence, or otherwise, find
2. \(\mathrm { P } ( A \cup B )\),
3. \(\mathrm { P } \left( \begin{array} { l l } A & B ^ { \prime } \end{array} \right)\)

11 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.

A student is picked at random from the class.

Given that this student plays squash, find the probability that this student does not play hockey. Two different students are picked at random from the class, one after the other, without replacement.

Given that the first student plays squash, find the probability that the second student plays hockey.

1 Four children, A, B, C and D, discuss how many of the 23 birthday parties held by their classmates they had gone to. Each party was attended by at least one of the four children. The results are shown in the Venn diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{50697293-6cdc-475f-981f-71a351b0ff5a-2_387_618_589_246}

1. Construct a complete graph \(\mathrm { K } _ { 4 }\), with vertices representing the four children and arcs weighted to show the number of parties each pair of children went to.
2. State a piece of information about the number of parties the children went to that is shown in the Venn diagram but is not shown in the graph. A fifth child, E, also went to some of the 23 parties shown in the Venn diagram.
   Every party that E went to was also attended by at least one of \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D .
   • A was at 8 of these parties, B at 7, C at 5 and D at 8 .
   • These include 5 parties attended by both A and \(\mathrm { B } , 2\) by both A and \(\mathrm { C } , 3\) by both A and \(\mathrm { D } , 3\) by both B and D and 4 by both C and D .
   • These include 1 party attended by \(\mathrm { A } , \mathrm { B }\) and D and 1 party attended by \(\mathrm { A } , \mathrm { C }\) and D .
   • Use the inclusion-exclusion principle to determine the number of parties that E went to.

3 Events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = 0.6 , \mathrm { P } ( B ) = 0.4\) and \(\mathrm { P } ( A \cup B ) = 0.8\).

1. Find \(\mathrm { P } ( A \cap B )\).
2. Find \(\mathrm { P } \left( A \cap B ^ { \prime } \right)\).
3. Find \(\mathrm { P } ( A \mid B )\). Events \(A\) and \(B\) are as above and a third event \(C\) is such that \(\mathrm { P } ( A \cup B \cup C ) = 1 , \mathrm { P } ( A \cap B \cap C ) = 0.05\), \(\mathrm { P } ( A \cap C ) = \mathrm { P } ( B \cap C )\) and \(\mathrm { P } \left( A \cap B ^ { \prime } \cap C ^ { \prime } \right) = 3 \mathrm { P } \left( A ^ { \prime } \cap B \cap C ^ { \prime } \right)\).
4. Find \(\mathrm { P } ( C )\).

1.
\includegraphics[max width=\textwidth, alt={}, center]{6dfefd72-338f-40be-ac37-aef56bfaccaa-02_399_743_248_662} The Venn diagram, where \(p\) is a probability, shows the 3 events \(A , B\) and \(C\) with their associated probabilities.

1. Find the value of \(p\).
2. Write down a pair of mutually exclusive events from \(A , B\) and \(C\).

3. In an after-school club, students can choose to take part in Art, Music, both or neither. There are 45 students that attend the after-school club. Of these

• 25 students take part in Art
• 12 students take part in both Art and Music
• the number of students that take part in Music is \(x\)
  1. Find the range of possible values of \(x\)

One of the 45 students is selected at random.
Event \(A\) is the event that the student selected takes part in Art.
Event \(M\) is the event that the student selected takes part in Music.

Determine whether or not it is possible for the events \(A\) and \(M\) to be independent.

1 Jane wants to travel from home to the local town. Jane can do this by train, by bus or by both train and bus.

1. Give an example of a problem that Jane could be answering that would give a construction problem. A website gives Jane all the possible buses and trains that she could use.
   Jane finds 7 possible ways to make the journey.
   • 2 of the 7 journeys involve travelling by train for at least part of the journey
   • 6 of the 7 journeys involve travelling by bus for at least part of the journey
   • Use the inclusion-exclusion principle to find how many of the 7 journeys involve travelling by both train and bus.