probability theory

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 $X$, the probability of $X$ to be in a range $[a,b]$ 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 $P$ of some event $E$, denoted $P(E)$, is usually defined in such a way that $P$ 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 $1$ 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 $100\%$, or $1$.

• 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 $1/8$.
  • Then the probability of $B$ given $A$ is $1/2$, since $A \cap B=\{HHH\}$ which has probability $1/8$ and $A=\{HHH,TTT\}$ which has probability $2/8$, and $\frac{1/8}{2/8}=\frac{1}{2}.$
  • The conditional probability $P(B|A)$ is not always equal to the unconditional probability $P(B)$.
  • 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.