Six Sigma and Beyond: Statistics and Probability, Volume III

  1. Equivalent to universal space U

    • Related to a random experiment

      • Uncertainty of results or outcomes

    • Consists of all possible outcomes or samples

    • Each output is a simple or elementary event, S i

    • Discrete, e.g., selecting a card or coin tosses

    • Continuous, e.g., diameter or weight of rods

    • Finite or infinite

  2. Events of sample space: A

    • Simple or elementary event is one (individual) outcome of sample space, S i .

    • An event A is a subset or grouping of simple events; e.g., event A of n simple events: A = {S 1 , ..., S n }

    • Elements of event A are samples with defined characteristics; e.g., the odd numbers tossed on dice.

If the outcome of an experiment S i is an element of A, then "the event A has occurred."

EXAMPLES OF SETS

Given: Random experiment in a twice-tossed coin.

  1. Find the sample space S of the four possible outcomes:

    Simple events: S 1 = TT, S 2 = HT, S 3 = HH, S 4 = TH

    Sample space: S = (S 1 , S 2 , S 3 , S 4 )

  2. Find the subset or event A defined as when: At least one head occurs.

    Event A = (HT, HH, TH) = (S 2 , S 3 , S 4 )

  3. Find the subset or event B defined as when: At least one tail occurs.

    Event B = (TT, HT, TH) = (S 1 , S 2 , S 4 )

  4. Find the subset or event C defined as when: At least one head (event A) AND one tail (event B) occur.

    Event C = (HT, TH) = (S 2 , S 4 )

    "AND" in set theory is an "intersection," meaning "both A and B."

    C

    ‰

    A ‹‚ B

     

    ‰

    (S 2 , S 4 )

  5. Find the subset or event D defined as when: The first toss is a head.

    Event D = (HT, HH) = (S 2 , S 3 )

  6. Find the subset or event E defined as when: At least one head (event A) OR at least one tail (event B) occurs.

    Event E = (TT, HT, HH, TH) = (S 1 , S 2 , S 3 , S 4 )

    "OR" in set theory is "union," meaning "either A or B or both."

    E ‰ A ‹ƒ B = (S 1 , S 2 , S 3 , S 4 ) = S

  7. Find the event F that CANNOT occur assuming the event A does occur. Event A is defined as when at least one head occurs.

    Event A = (HT, HH, TH) = (S 2 , S 3 , S 4 )

    Event F is the event "not A."

    Event F = (TT) = (S 1 )

    Event A and event F are " mutually exclusive" or " disjoint " in set theory.

    A ‹‚ F = =A ‹‚ A'

    Hence the event

    F = A' = (S 1 ) = (TT)

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