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:

  • 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.
  • Independence

    • In probability theory, to say that two events are independent means that the occurrence of one does not affect the probability that the other will occur.
    • For independent events, the condition does not change the probability for the event.
    • The third statement says that the probability of both independent events $A$ and $B$ occurring is the same as the probability of $A$ occurring, multiplied by the probability of $B$ occurring.
    • What is the probability that the coin will land on heads again?
    • Therefore, the probability of getting tails $4$ times in a row is:
  • The Multiplication Rule

    • The multiplication rule states that the probability that $A$ and $B$ both occur is equal to the probability that $B$ occurs times the conditional probability that $A$ occurs given that $B$ occurs.
    • In probability theory, the Multiplication Rule states that the probability that $A$ and $B$ occur is equal to the probability that $A$ occurs times the conditional probability that $B$ occurs, given that we know $A$ has already occurred.
    • We obtain the general multiplication rule by multiplying both sides of the definition of conditional probability by the denominator.
    • The probability that we get a $2$ on the die and a tails on the coin is $\frac{1}{6}\cdot \frac{1}{2} = \frac{1}{12}$, since the two events are independent.
    • Apply the multiplication rule to calculate the probability of both $A$ and $B$ occurring
  • Expected Values of Discrete Random Variables

    • The probability distribution of a discrete random variable $X$ lists the values and their probabilities, such that $x_i$ has a probability of $p_i$.
    • The probabilities $p_i$ must satisfy two requirements:
    • The sum of the probabilities is 1: $p_1+p_2+\dots + p_i = 1$.
    • In probability theory, the expected value (or expectation, mathematical expectation, EV, mean, or first moment) of a random variable is the weighted average of all possible values that this random variable can take on.
    • Suppose random variable $X$ can take value $x_1$ with probability $p_1$, value $x_2$ with probability $p_2$, and so on, up to value $x_i$ with probability $p_i$.
  • 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.
  • The Binomial Formula

    • The probabilities of these possibilities are shown in Table 2 and in Figure 1.
    • Table 1 is a discrete probability distribution: It shows the probability for each of the values on the $x$-axis.
    • In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of $n$ independent yes/no experiments, each of which yields success with probability $p$.
    • The probability of getting exactly $k$ successes in $n$ trials is given by the Probability Mass Function:
    • Employ the probability mass function to determine the probability of success in a given amount of trials
  • 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.
  • 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:
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