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Probability - PowerPoint PPT Presentation

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Probability. Laws of Chance. Language of Uncertainty. “The scientific interpretation of chance begins when we introduce probability.” -- David Ruelle. Probability. The notion of chance has existed for centuries. Egyptian tombs from around 2000 B.C.

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  • Laws of Chance.

  • Language of Uncertainty.

    “The scientific interpretation of chance

    begins when we introduce probability.”

    -- David Ruelle


  • The notion of chance has existed for centuries.

    • Egyptian tombs from around 2000 B.C.

    • Card and Board games from 14th century

  • Probability Quantifies Uncertainty.

    • 0  P(A)  1

    • Interpret P(A)=0 and P(A)=1

  • Basis of Inferential Statistics

Classical definition of probability
Classical Definition of Probability

  • Let n be the total number of outcomes possible, and assume that all outcomes are equally likely.

  • Let m be the number of distinct outcomes that comprise the event A.

  • The probability of event A occurring is:

    P(A) = m / n

Theoretical probability
Theoretical Probability

  • The classical definition of probability provides the theoretical probability of event A. The theoretical probability is not always calculable.

    • Examples:

      • In some situations, it is not possible to count all outcomes.

      • The outcomes are not equally likely to occur in all situations.

Empirical probability
Empirical Probability

  • The empirical probability of an event is the observed relative frequency of occurrence of that event if the experiment is repeated many times.

  • The empirical probability converges to the theoretical probability (truth) as the number of repetitions gets large.

Probability terminology
Probability Terminology

  • Experiment

    • an activity resulting in an uncertain outcome

  • Sample Space (S)

    • set of all possible outcomes in an experiment

  • Event (A)

    • set of some of the possible outcomes of an experiment

      • Any event is a subset of the sample space

      • An event is said to occur if the outcome of the experiment is a member of it.

Probability notation
Probability Notation

  • P(A) – denotes the probability of event A occurring ( 0  P(A)  1 )

  • n(A) – denotes the number of distinct outcomes in event A

  • Classical Definition of Probability:

Complement of an event
Complement of an Event

  • The complement of event A (denoted A’) contains all elements in the sample space that are not in A.

    • A’ occurs when A does not occur.

  • Complement Rule:

    • Many problems are easier to solve using the complement.

Discrete probability distributions
Discrete Probability Distributions

  • A discrete probability distribution specifies the probability associated with each possible distinct value of the random variable.

  • A probability distribution can be expressed in the form of a graph, table or formula.

    • For example: Let X be the number of heads that you get when you flip 2 fair coins.

Probability function
Probability Function

  • A probability function, denoted P(x), assigns probability to each outcome of a discrete random variable X.

  • Properties:

Binomial probability distribution
Binomial Probability Distribution

  • Results from an experiment in which a trial with two possible outcomes is repeated n times.

    • Heads/Tails, Yes/No, For/Against, Cure/No Cure

  • One outcome is arbitrarily labeled a success and the other a failure

  • Assumptions:

    • n independent trials

    • Probability of success is p in each trial

      (so q=1-p is the probability of failure)

  • Binomial random variable
    Binomial Random Variable

    • Let X be the number of success in n trials, then X is a binomial random variable.

      • Often, p is defined to be the proportion of the population with a characteristic of interest, and X is the number sampled with that characteristic of interest.

    • Probability Function

    Binomial probability formula
    Binomial Probability Formula


    P(x) = • px•qn-x

    (n - x )! x!

    Probability of x successes among n trials for any one particular order

    Number of

    outcomes with exactly x successes among n trials

    Binomial mean var st dev
    Binomial Mean, Var. & St. Dev.

    • The mean, variance and standard deviation of a binomial random variable with n trials and probability of success p: