In a fourfold table showing binary agreement of two observers, the observed proportion of agreement, p0, can be paradoxically altered by the chance-corrected ratio that creates kappa as an index of concordance. In one paradox, a high value of p0 can be drastically lowered by a substantial imbalance in the table's marginal totals either vertically or horizontally. In the second pardox, kappa will be higher with an asymmetrical rather than symmetrical imbalanced in marginal totals, and with imperfect rather than perfect symmetry in the imbalance. An adjustment that substitutes kappa max for kappa does not repair either problem, and seems to make the second one worse.