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statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, the conditional probability table (CPT) is defined for a set of discrete and mutually
dependent A dependant is a person who relies on another as a primary source of income. A common-law spouse who is financially supported by their partner may also be included in this definition. In some jurisdictions, supporting a dependant may enabl ...
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s to display
conditional probabilities In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occur ...
of a single variable with respect to the others (i.e., the probability of each possible value of one variable if we know the values taken on by the other variables). For example, assume there are three random variables x_1,x_2, x_3 where each has K states. Then, the conditional probability table of x_1 provides the conditional probability values P(x_1=a_k\mid x_2,x_3) – where the vertical bar , means “given the values of” – for each of the ''K'' possible values a_k of the variable x_1 and for each possible combination of values of x_2,\, x_3. This table has K^3 cells. In general, for M variables x_1,x_2,\ldots,x_M with K_i states for each variable x_i, the CPT for any one of them has the number of cells equal to the product K_1K_2\cdots K_M. A conditional probability table can be put into
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
form. As an example with only two variables, the values of P(x_1=a_k\mid x_2=b_j)=T_, with ''k'' and ''j'' ranging over ''K'' values, create a ''K''×''K'' matrix. This matrix is a
stochastic matrix In mathematics, a stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability. It is also called a probability matrix, transition matrix, ...
since the columns sum to 1; i.e. \sum_k T_ = 1 for all ''j''. For example, suppose that two
binary variable Binary data is data whose unit can take on only two possible states. These are often labelled as 0 and 1 in accordance with the binary numeral system and Boolean algebra. Binary data occurs in many different technical and scientific fields, wher ...
s ''x'' and ''y'' have the
joint probability distribution Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered ...
given in this table: Each of the four central cells shows the probability of a particular combination of ''x'' and ''y'' values. The first column sum is the probability that ''x'' =0 and ''y'' equals any of the values it can have – that is, the column sum 6/9 is the
marginal probability In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables ...
that ''x''=0. If we want to find the probability that ''y''=0 ''given'' that ''x''=0, we compute the fraction of the probabilities in the ''x''=0 column that have the value ''y''=0, which is 4/9 ÷ 6/9 = 4/6. Likewise, in the same column we find that the probability that ''y''=1 given that ''x''=0 is 2/9 ÷ 6/9 = 2/6. In the same way, we can also find the conditional probabilities for ''y'' equalling 0 or 1 given that ''x''=1. Combining these pieces of information gives us this table of conditional probabilities for ''y'': With more than one conditioning variable, the table would still have one row for each potential value of the variable whose conditional probabilities are to be given, and there would be one column for each possible combination of values of the conditioning variables. Moreover, the number of columns in the table could be substantially expanded to display the probabilities of the variable of interest conditional on specific values of only some, rather than all, of the other variables.


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Table Table may refer to: * Table (furniture), a piece of furniture with a flat surface and one or more legs * Table (landform), a flat area of land * Table (information), a data arrangement with rows and columns * Table (database), how the table data ...