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Sample spaces, tables and Venn diagrams

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Sample Spaces

  • A sample space lists all possible outcomes of an experiment or combined experiment.
  • For a single event, list outcomes systematically, e.g. rolling a fair die: {1, 2, 3, 4, 5, 6}.
  • For two events, use a sample space diagram (a grid) to show every possible pair of outcomes.
  • Example: tossing two coins – outcomes: HH, HT, TH, TT.
  • A sample space can contain unequal-probability outcomes. Counting favourable outcomes over total works only when the listed outcomes are equally likely.
  • For equally likely elementary outcomes, probability = favourable outcomes ÷ total outcomes. Otherwise use their actual probabilities.

Sums of two independent fair dice

Sums of two independent fair diceSecond dieFirst die123456123456234567345678456789567891067891011789101112P(total = 7) = 6/36 = 1/6

Two-Way Tables

  • A two-way table organises outcomes of two events in rows and columns.
  • Each cell shows the combined outcome or its count.
  • To find a probability from a two-way table, read the relevant cell(s) and divide by the grand total.
  • Example: a table of students by gender and subject choice – the cell for 'female and Maths' gives that count.
  • Always check that row totals and column totals add up to the grand total.

Two-way survey table

Two-way survey tableMusicNo musicWalkBusTotalTotal1289112020211940

Venn Diagrams

  • A Venn diagram uses circles to represent sets inside a rectangle (the universal set, often denoted U).
  • Overlapping circles show the intersection – elements in both sets.
  • The union is all elements in either set or both.
  • Fill the intersection first, then work outwards to avoid double-counting.
  • Remember to include elements that are in U but outside all circles – they belong to neither set.

Intersection of A and B

Intersection of A and BℰAB71058A ∩ Bn(ℰ) = 30

Set Notation

  • A \cap B means the intersection of sets A and B – elements in both.
  • A \cup B means the union – elements in A or B or both.
  • A' (or \bar{A}) means the complement of A – elements not in A.
  • \emptyset is the empty set – no elements.
  • n(A) means the number of elements in set A.
  • Do not confuse \cap (intersection) with \cup (union) – remember \cap looks like an 'n' for 'and'.

Complement inside the universal set

Complement inside the universal setℰAB71058A′n(ℰ) = 30

Calculating Probabilities from Venn Diagrams

  • Once a Venn diagram is complete, probabilities can be read directly.
  • P(A \cap B) = \frac{n(A \cap B)}{n(U)}.
  • P(A \cup B) = \frac{n(A \cup B)}{n(U)}.
  • P(A') = 1 - P(A).
  • For 'neither A nor B', use the region outside both circles: P((A \cup B)') = 1 - P(A \cup B).

Probability of a union

Probability of a unionℰAB1510510A ∪ Bn(ℰ) = 40

Common Misconceptions

  • Double-counting the intersection when finding a union – subtract n(A \cap B) once.
  • Starting with the known overlap is helpful when set totals are given. Other mathematically valid solution orders are possible.
  • Forgetting elements outside all sets but inside the universal set – they are part of U and must be included.
  • Confusing \cap and \cup symbols – check the meaning.
  • Listing outcomes unsystematically can miss cases – use a table or grid to be systematic.

Typical Question Forms

  • Complete a two-way table from given information and find a probability.
  • Complete a Venn diagram from given information (e.g., totals and intersection).
  • Find a probability using set notation like P(A \cup B) or P(A').
  • List all outcomes for two combined events (e.g., rolling a die and tossing a coin).
  • Find how many elements satisfy neither condition – use the complement region.

Slide

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Soal latihan

Pratinjau gratis — 8 dari 61 soal. Daftar untuk melihat semuanya.

  1. 1.Which symbol represents the intersection of two sets?

    Easy
    • A∩
    • B∪
    • C⊆
    • D∅

  2. 2.Which symbol represents the union of two sets?

    Easy
    • A∪
    • B∩
    • C⊂
    • D∈

  3. 3.In a Venn diagram, which region represents the complement of set A?

    Easy
    • AThe region inside A
    • BThe region outside A
    • CThe intersection of A and B
    • DThe union of A and B

  4. 4.When set totals and the overlap count are given, filling the intersection first is a useful strategy for completing a Venn diagram.

    Easy

    True or false?

  5. 5.A fair six-sided dice is rolled twice. How many outcomes are in the sample space?

    Medium
    • A6
    • B12
    • C36
    • D216

  6. 6.Which of the following are correct ways to list the sample space for tossing two coins? (Select all that apply)

    Medium
    • AHH, HT, TH, TT
    • BHH, HT, TT
    • CH, T
    • DHH, TH, TT, HT

  7. 7.Match each set notation to its meaning.

    Easy
    • A ∩ B
    • A ∪ B
    • A'
    • Elements in both A and B
    • Elements in A or B (or both)
    • Elements not in A

  8. 8.Put the steps for completing a Venn diagram from given information in the correct order.

    Medium
    • Fill the intersection
    • Fill the 'only' parts
    • Fill the 'neither' region
    • Check totals

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