# Probability and Counting Rules

In probability theory and statistics, a probability distribution is a way of describing the probability of an event, or the possible outcomes of an experiment, in a given state of the world. The set of all possible outcomes of the experiment (the sample space) is a subset of the sample space of all possible states of the world (the state space). For example, the probability that a fair coin comes up heads is 0.5, or half of the 2 possible outcomes of a coin flip (heads or tails). In the state space, the event "heads" corresponds to the outcome "heads" (followed by a coin flip), and "tails" corresponds to the outcome "tails" (followed by a coin flip). Similarly, the event "tail" corresponds to the outcome "tails" (followed by a coin flip), and "heads" corresponds to the outcome "heads" (followed by a coin flip). In the state space, "tails" and "heads" are the two possible outcomes of a coin flip, and the event of "tails" corresponds to the outcome "tails" (followed by a coin flip), and the event of "heads" corresponds to the outcome "heads" (followed by a coin flip). In probability theory and statistics, the probability distribution of a random variable (a function whose domain is the sample space) is a table that lists the probabilities of the values of the random variable. The probability distribution of a discrete random variable, such as the number of heads in a coin toss, is a table that lists the probabilities of the outcomes in the sample space, with each outcome in the sample space as the row label and the outcome of the random variable as the column label. In probability theory and statistics, the probability distribution of a continuous random variable is a table that lists the probabilities of the values of the random variable on a range of values; for example, the probability distribution of a continuous random variable X with a probability density function "f"("x") is a table that lists the probabilities of the values of X on the interval [0, 1] . In probability theory and statistics, a probability density function (also called a probability function or probability density) is a function that describes the probability of the value of another random variable, called the "dependent" variable, given the value of the random variable describing the probability, called the "independent" variable. For example, the probability density function of the random variable X is the function that gives the probability that X is in a given interval (the interval "I") when the value of X is given by a random variable Y. For example, if the probability of X is 0.5, and the probability of Y is 0.5, then the probability density function of X is 0.5 at x = 0 and 0.5 at x = 1. The probability distribution of the random variable X is its probability density function; that is, its probability distribution is the function F(x) = P(X = x). Similarly, the probability distribution of the random variable Y is the function G(y) = P(Y = y). The probability distribution of X is the function F(x) to which G(y) is the inverse function. The probabilities of the outcomes of an experiment are often given as the frequencies of the outcomes in the sample space. For example, the probability that a fair coin comes up heads is half the probability that a coin will land heads up. The probabilities of the outcomes of an experiment are often given as the relative frequencies of the outcomes in the sample space. For example, the probability that a fair coin comes up heads is twice the frequency of a coin landing heads up. In probability theory and statistics, the probability distribution of a random variable is the probability density function of the random variable. The probability distribution of a discrete random variable is the probability table of the random variable. The probability distribution of a continuous random variable is the probability function of the random variable. In probability theory and statistics, a random variable or stochastic variable is a function whose domain is the sample space of a discrete or continuous random experiment. For example, the sample space of a fair coin toss is the set of all possible outcomes, which can be described by the following three-element set: "H" = {0, 1, 2}. The probability distribution of a random variable is the probability table of the random variable. The probability distribution of a random variable is the probability mass function of the random variable. The probability mass function is the probability distribution of a random variable given its probability distribution. The probability mass function is often denoted by the Greek letter ? (pi), the first letter of the word probability. For example, the sample space of a fair coin toss is the three-element set {