How do we calculate the exact probability of a partly specified table?
The following procedure is designed only for tables that are filled in from left to right and from top to bottom, like writing a paragraph. The empty rows are all at the bottom, and the empty cells in the partly completed row are all to the right. First, complete any row or column that has only one cell unspecified: this can be done in only one way that is compatible with the marginal totals. Then, starting from the partly completed contingency table T0, construct two complete contingency tables T1 and T2 as follows. T1 has the same column totals as T0 and comprises all the complete rows of T0, plus one complementary row, which can be filled in in only one way that is compatible with the column totals. Construct T2 from the partly filled row of T0 plus one complementary row. The column totals of T2 are equal to the contents of the last row of T1. At this stage, the rationale of the calculation begins to be clear. The hypothetical allocation of numbers in all possible ways to complete t
