Principle of Inclusion/Exclusion

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.

2 A survey is being carried out into the sports viewing habits of people in a particular area. As part of the survey, 250 people are asked which of the following sports they have watched on television in the past month.

• Football
• Cycling
• Rugby

The numbers of people who have watched these sports are shown in the Venn diagram. \includegraphics[max width=\textwidth, alt={}, center]{6015ae6c-bf76-4a0c-af0f-5c53f9c5ed2a-2_723_917_1183_575} One of the people is selected at random.

1. Find the probability that this person has in the past month
   (A) watched cycling but not football,
   (B) watched either one or two of the three sports.
2. Given that this person has watched cycling, find the probability that this person has not watched football.

1. A large college produces three magazines.

One magazine is about green issues, one is about equality and one is about sports.
A student at the college is selected at random and the events \(G , E\) and \(S\) are defined as follows \(G\) is the event that the student reads the magazine about green issues \(E\) is the event that the student reads the magazine about equality \(S\) is the event that the student reads the magazine about sports
The Venn diagram, where \(p , q , r\) and \(t\) are probabilities, gives the probability for each subset. \includegraphics[max width=\textwidth, alt={}, center]{10736735-3050-43eb-9e76-011ca6fa48b8-10_508_862_756_603}

1. Find the proportion of students in the college who read exactly one of these magazines. No students read all three magazines and \(\mathrm { P } ( G ) = 0.25\)
2. Find
   1. the value of \(p\)
   2. the value of \(q\) Given that \(\mathrm { P } ( S \mid E ) = \frac { 5 } { 12 }\)
3. find
   1. the value of \(r\)
   2. the value of \(t\)
4. Determine whether or not the events ( \(S \cap E ^ { \prime }\) ) and \(G\) are independent. Show your working clearly. \section*{Question 4 continued.} \section*{Question 4 continued.} \section*{Question 4 continued.}

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.

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

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.

In lines 46 and 47, the text says "Of the 11 people with unauthorised transactions, 3 could explain them as breaches of card security (typically losing the card) but 9 could not ... ." Place numbers in the three regions of the diagram consistent with the information in this sentence. [2] \includegraphics{figure_3}

There are 180 students at a college following a general course in computing. Students on this course can choose to take up to three extra options. 112 take systems support, 70 take developing software, 81 take networking, 35 take developing software and systems support, 28 take networking and developing software, 40 take systems support and networking, 4 take all three extra options.

1. In the space below, draw a Venn diagram to represent this information. [5]

A student from the course is chosen at random. Find the probability that this student takes

1. none of the three extra options, [1]
2. networking only. [1]

Students who want to become technicians take systems support and networking. Given that a randomly chosen student wants to become a technician,

1. find the probability that this student takes all three extra options. [2]

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. 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. 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. \(\mathrm { P } \left( \begin{array} { l l } A & B ^ { \prime } \end{array} \right)\)

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.

The shaded region on one of the Venn diagrams below represents \((A \cup C) \cap B\) Identify this Venn diagram. Tick (\(\checkmark\)) one box. [1 mark] \includegraphics{figure_13}

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

70% of the households in a town have a CD player and 45% have both a CD player and a personal computer (PC). 18% have neither a CD player nor a PC.

1. Illustrate this information using a Venn diagram. [3 marks]
2. Find the percentage of the households that do not have a PC. [2 marks]
3. Find the probability that a household chosen at random has a CD player or a PC but not both. [2 marks]

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.