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Instatistics, theconditional probability table (CPT) is defined for a set of discrete and mutuallydependentrandom variables to displayconditional probabilities 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${\displaystyle x_1,x_2,x_3}$ where each has${\displaystyle K}$ states. Then, the conditional probability table of${\displaystyle x_1}$ provides the conditional probability values${\displaystyle P(x_1=a_k\mid x_2,x_3)}$ – where the vertical bar${\displaystyle |}$ means “given the values of” – for each of theK possible values${\displaystyle a_k}$ of the variable${\displaystyle x_1}$ and for each possible combination of values of${\displaystyle x_2,x_3.}$ This table has${\displaystyle K^3}$ cells. In general, for${\displaystyle M}$ variables${\displaystyle x_1,x_2,\dots ,x_M}$ with${\displaystyle K_i}$ states for each variable${\displaystyle x_i,}$ the CPT for any one of them has the number of cells equal to the product${\displaystyle K_1{K}_2\cdots K_M.}$^[1]

A conditional probability table can be put intomatrix form. As an example with only two variables, the values of${\displaystyle P(x_1=a_k\mid x_2=b_j)=T_{kj},}$ withk andj ranging overK values, create aK×K matrix. This matrix is astochastic matrix since the columns sum to 1; i.e.${\displaystyle \sum _k{T}_{kj}=1}$ for allj. For example, suppose that twobinary variablesx andy have thejoint probability distribution given in this table:

Each of the four central cells shows the probability of a particular combination ofx andy values. The first column sum is the probability thatx =0 andy equals any of the values it can have – that is, the column sum 6/9 is themarginal probability thatx=0. If we want to find the probability thaty=0given thatx=0, we compute the fraction of the probabilities in thex=0 column that have the valuey=0, which is 4/9 ÷ 6/9 = 4/6. Likewise, in the same column we find that the probability thaty=1 given thatx=0 is 2/9 ÷ 6/9 = 2/6. In the same way, we can also find the conditional probabilities fory equalling 0 or 1 given thatx=1. Combining these pieces of information gives us this table of conditional probabilities fory:

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.

• Conditional probability

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