The author tried to calculate the joint probability of a square to dance a sequence correctly from the probabilities of the individual dancers. The probability of a square composed of eight dancers each with a 90% probability of dancing error-free is given as .9 x .9 x .9 x .9 x .9 x .9 x .9 x.9 = .4305 = 43%
However, this formula to calculate joint probabilities from individual probabilities is only correct if the individual events are independent. The formula is applicable for eight dancers in different squares (if possible in different halls), but not for eight dancers in the same square, in which the dancers (let us hope) are not dancing independently from each other.
How can we calculate the joint probability of the square from the individual probabilities? Not at all. There are assumptions required for the pattern of dependencies in the square. However, there are so many factors influencing the dependencies in a square (properties of the dancers, caller, mood in the hall, ...) that it is difficult to formulate an appropriate (statistical) model. Although theoretically possible, such a model would be very complicated and nevertheless only an insufficient description of the truth. The dependencies in a square (and other things of life) are quite complex and variable and cannot be adequately quantified by simple numbers. So I won't even try that.
The dependencies within a square were taken into account in the article. However, the method used was grossly false (from the sight of probability calculation). On page 84 and following, the dancers who can correct the errors of other dancers get a "probability" of more than 100%. This is nonsense or at least has nothing to do with probability calculation. A probability always lies between 0 and 100% (limits included), or it is no probability. (For experts: This follows from the axioms of Kolmogorow.) All the following calculations and numbers are
based on this false concept and have no meaning at all.
I underline that my critique only addresses the meaningless probability calculations and not the main statements of the article. My tip: If someone has good arguments for any point of view, then one should present these arguments. An assistance of the arguments by means of statistics is often unnecessary (and bores most of the readers). However, if statistics and probability should be used then consult someone who understands the basic rules of probability calculation. There is the risk that good arguments are weakened by the use of false and meaningless numbers.