To see the efficiency of the statistical methods for positive identification of fathers using genetic markers, the statistical properties of the paternity index are studied algebraically and numerically. It is found that the currently used statistical methods are not powerful enough to discriminate between true fathers and non-excluded non-fathers, and, more often than not, may lead to false attributions of paternity. It is, therefore, suggested that exclusion of paternity is the only conclusive evidence that can be accepted by courts of law until better methods are devised.