Simple Probability Diagrams (AQA GCSE Maths)

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  • True or False?

    A two-way table is used to compare two types of characteristics.

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  • True or False?

    A two-way table is used to compare two types of characteristics.

    True.

    A two-way table is used to compare two types of characteristics.

    E.g. school year group and favourite genre of movie.

  • How do you construct a two-way table from information given in words?

    1. Identify the two characteristics, e.g. favourite colours, gender

    2. Use rows for one characteristic and columns for the other

    3. Add an extra row and column for marginal totals

    Red

    Blue

    Yellow

    Total

    Male

    Female

    Total

  • True or false?

    The numbers needed to complete a two-way table will always be given explicitly in a question.

    False.

    When completing a two-way table, some values can be filled in directly from the question information, but some values will need to be worked out.

    E.g. you may need to subtract other values in a row from the row total to find a missing value.

  • How can you double-check your answers when completing a two-way table?

    You can double-check your answers when completing a two-way table by making sure that all row and column totals add up correctly, and that they match the grand total.

  • How can the probability of an event occurring be worked out from a two-way table?

    E.g. what is the probability that a randomly selected student's favourite subject is Physics?

    Biology

    Physics

    Chemistry

    Total

    Year 7

    12

    8

    10

    30

    Year 8

    8

    13

    6

    27

    Total

    20

    21

    16

    57

    The probability of a particular event occurring can be worked out by finding the number of successes by the total number.

    E.g. the probability that a student's favourite subject is Physics is 21 over 57.

    Biology

    Physics

    Chemistry

    Total

    Year 7

    12

    8

    10

    30

    Year 8

    8

    13

    6

    27

    Total

    20

    21

    16

    57

  • What is a frequency tree?

    A frequency tree is a diagram that shows the frequencies associated with two characteristics of a set of data.

    E.g. The below frequency tree shows the number of ducks and swans, broken down by male and female.

    frequency tree showing number of ducks and swans, broken down by male and female
  • True or False?

    Frequency trees are usually used when each characteristic has only two possible outcomes of interest.

    True.

    Frequency trees are usually used when each characteristic has only two possible outcomes of interest.

  • What is the purpose of the 'bubble' at the start of a frequency tree?

    The bubble at the start of a frequency tree contains the total frequency of all outcomes.

  • True or False?

    The order of characteristics on the branches of a frequency tree matters.

    False.

    The order of characteristics on the branches of a frequency tree does not matter.

    Strictly speaking, it does not matter which set of branches has which characteristic.

  • True or False?

    Frequencies in a frequency tree always increase from left to right.

    False.

    Frequencies in a frequency tree do not always increase from left to right.

    In general, the values decrease from left to right as the total frequency is broken down in a frequency tree.

  • What is a quick check to ensure a frequency tree is completed correctly?

    A quick check to ensure a frequency tree is completed correctly is to make sure the values in the bubbles at the end of each set of branches add up to the total frequency.

  • What does the first set of branches in a frequency tree represent?

    The first set of branches in a frequency tree breaks down the total frequency into the frequencies for the outcomes of the first characteristic.

  • True or False?

    Frequency trees can easily handle three or more characteristics.

    False.

    Frequency trees can not easily handle more than three characteristics

    While it is possible to have three or more characteristics in a frequency tree, such diagrams would quickly become large and cumbersome.

  • What is a set?

    A set is a collection of elements.

  • What symbol is the empty set represented by?

    represents the empty set (the set with no elements).

  • What notation is used to represent the number of elements in a set?

    E.g. the number of elements in set A.

    n(A) represents the number of elements in set A.

  • What region in the Venn diagram represents the union of A and B, open parentheses A union B close parentheses?

    A Venn diagram with a rectangle containing two circles, sets A and B.

    The union of A and B, open parentheses straight A union straight B close parentheses, is the set of all elements in A or B or both.

    A Venn diagram with two overlapping circles labelled A and B inside a rectangle labelled U, representing a universal set. Both circles are shaded.
  • Which region represents A' on the Venn diagram below?

    A Venn diagram with a rectangle containing two circles, sets A and B.

    A' means not A, (A' is the complement of A) .
    The regions not in the A circle, (everything outside A), represent A'.

    A Venn diagram with two overlapping circles labelled A and B on a blue rectangle, indicating the region not in set A.
  • True or False?

    The region shaded below represents the intersection of A and B, open parentheses A intersection B close parentheses.

    Venn diagram with three overlapping circles labelled A, B, and C within a rectangle labeled U. The middle regions, where all three circles overlap, is shaded.

    False.

    The region shown is the intersection of A, B and C, open parentheses straight A intersection straight B intersection straight C close parentheses.

    The intersection of A and B, open parentheses straight A intersection straight B close parentheses, is:

    A Venn diagram with three intersecting circles labelled A, B, and C within a rectangle labelled U. The intersection of circles A and B is shaded.
  • What notation is used when writing out a set of elements?

    E.g. set A contains the elements 1, 2 and 3.

    A set containing a number of elements is shown with the elements written inside curly brackets {}.

    E.g. A = {1, 2, 3}.

  • True or False?

    The universal set is represented by the rectangle in a Venn diagram.

    True.

    The universal set is represented by the rectangle in a Venn diagram.

  • What does the symbol represent?

    is the universal set (the set of everything).

    You may also see other symbols for the universal set, such as U, S or xi.

  • What does a element of A mean?

    a element of straight A means a is an element of A.

    a is in the set A, or a is a member of the set A.

  • Describe how to find straight P open parentheses A close parentheses from a Venn diagram that shows sets A and B.

    E.g. find straight P open parentheses A close parentheses from the Venn diagram.

    Venn diagram with sets A and B. Set A only has 8, the intersection of sets A and B has 2 and set B only has 15. Within the universal set, U, but outside both sets A and B is 7.

    straight P open parentheses A close parentheses is the probability of being in set A.

    This is the number inside the full circle of set A divided by the total number of the whole Venn diagram, e.g. straight P open parentheses A close parentheses equals fraction numerator 8 plus 2 over denominator 8 plus 2 plus 15 plus 7 end fraction equals 10 over 32.

    Venn diagram with two intersecting circles labelled A and B. Set A only contains 8 the intersection contains 2 and set B only contains 15. The number 7 lies within the universal set, U, but outside both sets A and B. The whole of set A is highlighted green.
  • Describe how to find straight P open parentheses A intersection B close parentheses from a Venn diagram that shows sets A and B.

    E.g. find straight P open parentheses A intersection B close parentheses from the Venn diagram.

    Venn diagram with sets A and B. Set A only has 8, the intersection of sets A and B has 2 and set B only has 15. Within the universal set, U, but outside both sets A and B is 7.

    straight P open parentheses A intersection B close parentheses is the probability of being in the intersection of set A and set B.

    This is the value inside the overlapping region of set A and set B divided by the total number of the whole Venn diagram, e.g. straight P open parentheses A intersection B close parentheses equals fraction numerator 2 over denominator 8 plus 2 plus 15 plus 7 end fraction equals 2 over 32.

    Venn diagram with two intersecting circles labelled A and B. Set A only contains 8 the intersection contains 2 and set B only contains 15. The number 7 lies within the universal set, U, but outside both sets A and B. The intersection of set A and B is highlighted.
  • True or False?

    If A and B are mutually exclusive, then straight P open parentheses A intersection B close parentheses equals 0.

    True.

    If A and B are mutually exclusive, then straight P open parentheses A intersection B close parentheses equals 0.

    On a Venn diagram, if A and B are mutually exclusive then their circles do not overlap (they cannot both happen at the same time).

    This makes being in the intersection impossible, so straight P open parentheses A intersection B close parentheses equals 0.

  • True or False?

    To find straight P open parentheses A union B close parentheses you need to double-count the numbers in the intersection (overlap) as they occur twice.

    False.

    To find straight P open parentheses A union B close parentheses you do not double-count the numbers in the intersection A intersection B, you just count them once.

  • Describe which region on a Venn diagram is required to calculate straight P open parentheses A intersection B intersection C close parentheses.

    E.g. find straight P open parentheses A intersection B intersection C close parentheses from the Venn diagram.

    Venn diagram with three overlapping circles labelLed A, B, and C. Regions contain numbers 5, 2, 11, 6, 3, 1, 9, and 8. U is the universal set.

    The region required to calculate straight P open parentheses A intersection B intersection C close parentheses is the one that is the overlap of all three sets A, B and C. E.g. straight P open parentheses A intersection B intersection C close parentheses equals fraction numerator 3 over denominator 5 plus 2 plus 11 plus 6 plus 3 plus 1 plus 9 plus 8 end fraction equals 3 over 45.

    Venn diagram with three circles labelled A, B, and C. Numbers inside segments: A-5, B-11, C-9, AB-2, AC-6, BC-1, ABC-3. Inside the universal set but outside sets A, B and C is the number 8. The intersection between all three circles is highlighted.
  • True or False?

    On a Venn diagram showing sets A and B, the region required to calculate straight P open parentheses A apostrophe close parentheses is the part of set B that does not overlap A.

    False.

    On a Venn diagram showing sets A and B, the region required to calculate straight P open parentheses A apostrophe close parentheses is anything that is outside the circle of A, e.g.

    Venn diagram with two intersecting circles labelled A and B. Set A only contains 8 the intersection contains 2 and set B only contains 15. The number 7 lies within the universal set, U, but outside both sets A and B. The whole of the universal set, except for set A, is highlighted green.
  • On a Venn diagram showing sets A and B, explain how to calculate straight P open parentheses A vertical line B close parentheses.

    E.g. find