Sample spaces and events



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  • The following topics will be covered in this lecture:

    • Random experiments
    • Discrete and Continuous Sample Spaces
    • Events and set notation
    • Permutations and combinatinos

Random experiments

  • Suppose we conducting an experiment: measure the current in a thin copper wire.
  • The measurements can differ slightly because of small variations in variables that are not controlled in our experiment, e.g.:
    • changes in ambient temperatures;
    • slight variations in the gauge;
    • impurities in the chemical composition of the wire;
    • current source drifts; and/or
    • measurement errors.
  • No matter how carefully an experiment is designed and conducted, the variation is almost always present
Noise variables affect the transformation of inputs to outputs

Courtesy of Montgomery & Runger, Applied Statistics and Probability for Engineers, 7th edition

    Our goal is to understand, quantify, and model the type of variations that we often encounter.

  • Ours models incorporate uncontrollable inputs (noise) that combine with the controllable inputs to produce the output of our system.
  • Because of the uncontrollable inputs, the same settings for the controllable inputs do not result in identical outputs every time the system is measured.

Random experiments

    Random experiment: an experiment that can result in different outcomes, even though it is repeated in the same manner every time.
  • Example: in the design of a communication system, such as a voice communication network, the information capacity available to serve individuals using the network is an important design consideration.
  • For voice communication, sufficient external lines need to be available to meet the requirements of a business.
    • Assuming each line can carry only a single conversation, how many lines should be purchased?
  • If too few lines are purchased, calls can be delayed or lost.
  • The purchase of too many lines increases costs.
  • Increasingly, design and product development is required to meet customer requirements at a competitive cost.
  • In the design of the voice communication system, a model is needed for the number of calls and the duration of calls.
  • Even knowing that, on average, calls occur every five minutes and that they last five minutes is not sufficient.
    • If calls arrived precisely at five-minute intervals and lasted for precisely five minutes, one phone line would be sufficient.
    • However, the slightest variation in call number or duration would result in some calls being blocked by others.
  • A system designed without considering variation will be woefully inadequate for practical use.
  • Our model for the number and duration of calls needs to include variation as an integral component.

Sample spaces

The set of all possible outcomes of a random experiment is called the sample space of the experiment. The sample space is denoted as \( S \).
  • Consider an experiment that selects a cell phone camera and records the recycle time of a flash (the time taken to ready the camera for another flash).

  • Because the time is positive, it is convenient to define the sample space as simply the positive real line \[ S=R^+ = \{x|\: x>0 \} \]

  • If it is known that all recycle times are between 1.5 and 5 seconds, the sample space can be \[ S = \{x|\: 1.5 < x < 5 \} \]

  • If the objective of the analysis is to consider only whether the recycle time is low, medium, or high \[ S=\{low,\:medium,\:high \} \]

  • If the objective is only to evaluate whether or not a particular camera conforms to a minimum recycle-time specification \[ S=\{yes,\:no \} \]

  • The sample space \( S \) depends on the kinds of measurements we are taking as above.

Discrete and Continuous Sample Spaces

  • A sample space is continuous if it contains an interval (either finite or infinite) of real numbers.

    • Recall the camera flash example, \( S = R^+ \) is an example of a continuous sample space.
    • The unit of measurement can be arbitrarily sub-divided (seconds, miliseconds, etc.) and the measurement still makes sense.
  • A sample space is discrete if it consists of a finite or countable infinite set of outcomes.

    • Recall the camera flash example, \( S = \{yes,\: no\} \) is a discrete sample space.
    • The unit of measurement above cannot be arbitrarily sub-divided making the sample space discrete.
  • The best choice of a sample space depends on the objectives of the study.

Discrete and Continuous Sample Spaces

  • Suppose that the recycle times of two cameras are recorded.

  • The extension of the positive real line \( R \) is to take the sample space to be the positive quadrant of the plane \[ S=R^+ \times R^+ \]

  • Measurements would come in pairs of time units, one piece of data for each camera.

  • If the objective of the analysis is to consider only whether or not the cameras conform to the manufacturing specifications, either camera may:

    • (yes=y) conform; or
    • may not conform (no=n)
  • Then the sample space can be represented by the four outcomes: \[ S=\{yy,yn,ny,nn\} \]

  • If we are interested only in the number of conforming cameras in the sample, then the sample space can be \[ S=\{0,1,2\} \]

Tree diagrams

  • Consider an experiment in which cameras are tested until the flash recycle time fails to meet the specifications. The sample space can be represented as \[ S=\{n, yn, yyn, yyyn, yyyyn, \text{and so forth}\} \]
  • Sample spaces can also be described graphically with tree diagrams when a sample space can be constructed in several steps or stages.
  • Each of the \( n_1 \) ways of completing the first step as a branch of a tree.
  • Each of the ways of completing the second step can be represented as \( n_2 \) branches starting from the ends of the original branches, and so forth.
  • EXAMPLE: messages in a communication system are classified as to whether they are received within the time specified by the system design. The following a tree diagram represents the sample space of possible outcomes for three messages.
Tree diagram for three messages

Courtesy of Montgomery & Runger, Applied Statistics and Probability for Engineers, 7th edition

  • Each message can be received either on time or late
  • Stage 1: message 1
    • \( n_1=2 \) ways of completing message
  • Stage 2: message 2
    • \( n_2=2 \) ways of completing message
  • Stage 3: message 3
    • \( n_3=2 \) ways of completing message


  • An event is a subset of the sample space of a random experiment.
  • Events are subsets, we can use basic set operations
  • We can use Venn diagrams to represent a sample space and events in a sample space.
Union of two events is the event that consists of all outcomes that are contained in either of the two events.
Denoted \( A \cup B \) Venn diagram of two sets' union
Intersection of two events is the event that consists of all outcomes that are contained in both of the two events.
Denoted \( A \cap B \) Venn diagram of two sets' intersection
Complement of an event in a sample space is the set of outcomes in the sample space that are not in the event.
Denoted \( A' \) or \( A^c \) Venn diagram of complement set

Camera flash events

  • Recall the camera flash example with sample space \( S=\{yy,yn,ny,nn\} \).
  • Suppose that the subset of outcomes for which at least one camera conforms is denoted as \( E_1 \) \[ E_1 = \{yy, yn, ny\} \]
  • When the cameras do not conform, denote as event \( E_2 \), contains only the single outcome, \[ E_2 = \{nn\} \]
  • Other events are
    • \( E_3 = \emptyset \), the null set
    • \( E_4 = S \), the sample space
    • Question: If \( E_5 = \{yn, ny, nn\} \), what is …
      • \( E_1 \cup E_5 = \)?
        • \( E_1 \cup E_5 =S \)
      • \( E_1 \cap E_5 = \)?
        • \( E_1 \cap E_5 =\{yn,ny\} \)
      • \( E_1' = \)?
        • \( E_1' =\{nn\} \)

Hospital Emergency Events