Sample spaces, Venn diagrams and tree diagrams

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レッスンノート

Sample Spaces

  • A sample space is the set of all possible outcomes of an experiment or random trial.
  • It is usually written using set notation, with outcomes listed as elements inside curly brackets, e.g. S = {H, T} for a coin toss.
  • The sample space may be denoted by S, Ω, or U (for 'universal set').
  • The outcomes can be numbers, words, letters, or symbols.
  • A sample space can be finite, countably infinite, or uncountably infinite.
  • For a single six-sided die, the sample space is {1, 2, 3, 4, 5, 6}.
  • For two coins, the sample space is {HH, HT, TH, TT}, where the order matters (first coin, then second).

Events

  • An event is a subset of the sample space, usually denoted by E.
  • If the outcome of an experiment is in E, then event E has occurred.
  • For a single coin toss, possible events include E = {} (empty set), E = {H}, E = {T}, and E = {H, T}.
  • For two coins, the event 'at least one head' is E = {HH, HT, TH}.
  • Events can be shown visually: a rectangle represents the sample space, and ovals inside represent events.

Conditions for a Sample Space

  • Outcomes must be mutually exclusive: if one outcome occurs, no other outcome can occur at the same time.
  • Outcomes must be collectively exhaustive: on every trial, one of the outcomes in the sample space must occur.
  • A well-defined, non-empty sample space S is one component of a probabilistic model (a probability space).
  • The other components are a set of possible events and a probability assigned to each event.

Listing Outcomes Systematically

  • To find a sample space, list all possible outcomes in an organised way so none are missed.
  • For two events, use a sample space diagram (a grid) to show all combinations.
  • Example: rolling two dice, list pairs (1,1), (1,2), …, (6,6) — there are 36 outcomes.
  • Order matters when the events are distinguishable (e.g. first die and second die).

Venn Diagrams and Set Notation

  • A Venn diagram uses circles inside a rectangle to show sets and their relationships.
  • The rectangle represents the universal set (the sample space).
  • Union (A ∪ B) means 'A or B' — outcomes in A, in B, or in both.
  • Intersection (A ∩ B) means 'A and B' — outcomes in both A and B.
  • Complement (A') means 'not A' — outcomes in the universal set but not in A.
  • Venn diagrams help visualise events and their probabilities.

Mutually Exclusive Events and the Addition Rule

  • Mutually exclusive events cannot happen at the same time; their intersection is empty.
  • For mutually exclusive events A and B, P(A or B) = P(A) + P(B).
  • This is the addition rule for mutually exclusive events.
  • If events are not mutually exclusive, P(A or B) = P(A) + P(B) − P(A and B).

Independent Events and the Multiplication Rule

  • Independent events are events where the outcome of one does not affect the other.
  • For independent events A and B, P(A and B) = P(A) × P(B).
  • This is the multiplication rule for independent events.
  • Example: tossing two coins, P(two heads) = 1/2 × 1/2 = 1/4.

Tree Diagrams

  • A tree diagram shows all possible outcomes of two or more events in a branching structure.
  • Each branch represents a possible outcome, with its probability written on the branch.
  • To find the probability of a combination, multiply the probabilities along the branches.
  • The probabilities on branches from the same point should sum to 1.
  • Tree diagrams are useful for independent events and for events with two stages.

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練習問題

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  1. 1.What is the sample space of an experiment?

    Easy
    • AThe set of all possible outcomes of the experiment
    • BThe single outcome that actually happens
    • CThe list of outcomes that are impossible
    • DThe number of times the experiment is repeated

  2. 2.A single fair six-sided die is rolled once and the number of pips facing up is recorded. Which set is the sample space?

    Easy
    • A{1, 2, 3, 4, 5, 6}
    • B{0, 1, 2, 3, 4, 5, 6}
    • C{2, 4, 6}
    • D{1, 2, 3, 4, 5}

  3. 3.Two fair coins are tossed. Which set is the sample space, where H = heads and T = tails?

    Easy
    • A{HH, HT, TH, TT}
    • B{HH, TT}
    • C{H, T}
    • D{HH, HT, TT}

  4. 4.Two fair coins are tossed. The event E is that at least one of the coins is heads. Which set lists the outcomes in E?

    Medium
    • A{HH, HT, TH}
    • B{HH}
    • C{HT, TH}
    • D{HH, HT, TH, TT}

  5. 5.A sample space must satisfy certain conditions. Which of the following statements are true? (select all that apply)

    Medium
    • AThe outcomes must be mutually exclusive
    • BThe outcomes must be collectively exhaustive
    • CThe sample space must be empty
    • DEvery outcome must have the same probability

  6. 6.In probability, an event is a subset of the sample space.

    Easy

    True or false?

  7. 7.In a Venn diagram showing a sample space, how are the outcomes of the sample space usually represented?

    Medium
    • AAs points inside a rectangle
    • BAs points outside the rectangle
    • CAs arrows pointing to the rectangle
    • DAs labels written along the edge of the rectangle

  8. 8.In a Venn diagram, the outcomes of an event are shown by which region?

    Medium
    • AThe points inside the oval representing that event
    • BThe points outside the rectangle
    • CThe points on the boundary of the rectangle only
    • DThe points inside every other oval but not this one

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