Before we answer the question let us ensure that we have common understanding of what is positive or negative test cases. In HBT, positive test cases are those whose input values are valid, while a negative test case is one with at least one of the test inputs (i.e. Test data) being incorrect (i.e. out of specification).
The objective of positive test cases is “conformance” while the objective of negative test cases is “robustness”. If a majority of test case are positive, then it implies that we are primarily interested in conformance i.e. ensuring the system handles correct inputs well. This we know is not sufficient, as accidental incorrect inputs should not result in unexpected possibly risky/dangerous behaviour. Hence we need test cases that are indeed “negative”.
So, coming back to the question, what is a good enough distribution of positive and negative test cases? Any quick answer like 75% should not be trusted as they have no basis. So, how do we answer this question? Step back now and look at the number of inputs for a given test case, the clue is there. For example at a lower level of testing, where say we are validating a screen, the number of inputs may be many as the screen may be populated. As we go up the testing level, e.g. Testing a feature that uses a few screens, the test data is not the various individual inputs on the screen, but aggregate data (think like a record) and these may be fewer in number compared to the earlier levels.
Having understood that the number of test data (or inputs) at lower levels is far higher than those at the higher levels, it is only logical to conclude that the number of negative test cases at lower levels. Now how much should that be? To answer his finally without resorting to magic (!), let us illustrate with simple example. If there are 5 inputs and each input has six possible values (3 positive i.e. valid and 3 negative i.e. invalid) then using simple combinatorial math, we can see that there be (minimally) 3*5=15 negative test cases and (minimally) 5 positive test cases. In this case the 15/(15+5)= (75%) of test cases are negative.
In closing, understanding the number of inputs and clear understanding of what an input at a testing level is (The STEM Core concept “Input granularity principle” of HBT methodology helps in understanding as what an input at a level is) it is possible to quickly estimate the minimal number of negative test cases. This is very useful in quickly ascertaining whether the test cases are conformance and robustness oriented.