probability theory

(noun)

The mathematical study of probability (the likelihood of occurrence of random events in order to predict the behavior of defined systems).

Related Terms

  • independence

Examples of probability theory in the following topics:

  • Theoretical Probability

    • Probability theory uses logic and mathematical reasoning, rather than experimental data, to determine probable outcomes.
    • Mathematically, probability theory formulates incomplete knowledge pertaining to the likelihood of an event.
    • As such, the meteorologist's 60% verdict is a theoretical probability, and not the result of any proven experiment.
    • For example, the probability of rolling any specific number on a six-sided die is one out of six: there are six, equally probable sides to land on, and each side is distinct from the others.
    • This is a theoretical probability; testing by rolling the die many times and recording the results would result in an experimental probability.
  • Probability

    • Probability density function describes the relative likelihood, or probability, that a given variable will take on a value.
    • In probability theory, a probability density function (pdf), or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value.
    • The probability for the random variable to fall within a particular region is given by the integral of this variable's probability density over the region.
    • For a continuous random variable XXX, the probability of XXX to be in a range [a,b][a,b][a,b] is given as:
    • Apply the ideas of integration to probability functions used in statistics
  • Probability

    • Probability is a mathematical tool used to study randomness.
    • The expected theoretical probability of heads in any one toss is 1/2 or 0.5.
    • The theory of probability began with the study of games of chance such as poker.
    • Predictions take the form of probabilities.
    • You might use probability to decide to buy a lottery ticket or not.
  • Fundamentals of Probability

    • In probability theory, the probability PPP of some event EEE, denoted P(E)P(E)P(E), is usually defined in such a way that PPP satisfies a number of axioms, or rules.
    • Probability is a number.
    • If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities.
    • The probability that an event does not occur is 111 minus the probability that the event does occur.
    • The probability that an event occurs and the probability that it does not occur always add up to 100%100\%100%, or 111.
  • Remarks on the Concept of “Probability”

    • Inferential statistics is built on the foundation of probability theory, and has been remarkably successful in guiding opinion about the conclusions to be drawn from data.
    • Therefore the probability of heads is taken to be 1/2, as is the probability of tails.
    • Of course, wind direction also affects probability.
    • Questions such as "What is the probability that Ms.
    • An event with probability 0 has no chance of occurring; an event of probability 1 is certain to occur.
  • Conditional Probability

    • The conditional probability of an event is the probability that an event will occur given that another event has occurred.
    • Each individual outcome has probability 1/81/81/8.
    • Then the probability of BBB given AAA is 1/21/21/2, since A∩B={HHH}A \cap B=\{HHH\}A∩B={HHH} which has probability 1/81/81/8 and A={HHH,TTT}A=\{HHH,TTT\}A={HHH,TTT} which has probability 2/82/82/8, and 1/82/8=12.\frac{1/8}{2/8}=\frac{1}{2}.​2/8​​1/8​​=​2​​1​​.
    • The conditional probability P(B∣A)P(B|A)P(B∣A) is not always equal to the unconditional probability P(B)P(B)P(B).
    • In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule) is a result that is of importance in the mathematical manipulation of conditional probabilities.
  • Scientific Method

    • If a theory can accommodate all possible results then it is not a scientific theory.
    • Although strictly speaking, disconfirming an hypothesis deduced from a theory disconfirms the theory, it rarely leads to the abandonment of the theory.
    • Instead, the theory will probably be modified to accommodate the inconsistent finding.
    • This can lead to discontent with the theory and the search for a new theory.
    • If a new theory is developed that can explain the same facts in a more parsimonious way, then the new theory will eventually supersede the old theory.
  • Experimental Probabilities

    • It is a probability calculated from experience, not from theory.
    • Experimental probability contrasts theoretical probability, which is what we would expect to happen.
    • In statistical terms, the empirical probability is an estimate of a probability.
    • An advantage of estimating probabilities using empirical probabilities is that this procedure includes few assumptions.
    • A disadvantage in using empirical probabilities is that without theory to "make sense" of them, it's easy to draw incorrect conclusions.
  • Continuous Probability Distributions

    • A continuous probability distribution is a probability distribution that has a probability density function.
    • Each of these individual outcomes has probability zero, yet the probability that the outcome will fall into the interval (3 cm, 4 cm) is nonzero.
    • In theory, a probability density function is a function that describes the relative likelihood for a random variable to take on a given value.
    • Unlike a probability, a probability density function can take on values greater than one.
    • The standard normal distribution has probability density function:
  • Theoretical Perspectives in Sociology

    • Sociological theory is developed at multiple levels, ranging from grand theory to highly contextualized and specific micro-range theories.
    • Putnam's theory proposes:
    • This element of Putnam's theory clearly illustrates the basic purpose of sociological theory.
    • In short, Putnam's theory clearly encapsulates the key ideas of a sociological theory.
    • In fact, it is probably more useful and informative to view theories as complementary.
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