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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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probability
Probability
  • Laws of Chance.
  • Language of Uncertainty.

“The scientific interpretation of chance

begins when we introduce probability.”

-- David Ruelle

probability1
Probability
  • 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

n!

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:
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