Most states require that paternity be established by a "preponderance of the evidence," meaning that paternity (or lack of paternity) is "more likely than not." This test is the same test used in other civil matters, such as negligence, contract law and malpractice. We call this the 50.01% test, meaning that if evidence is even slightly greater than 50-50, it constitutes a "preponderance of evidence."
However, many states insist on a higher standard, known as the "clear and convincing evidence" test. This test is more stringent (harder to overcome) than the "preponderance" test, but not as strict as the "beyond a reasonable doubt" test, the standard in a criminal case.
Whatever the burden of proof on a person asserting or denying paternity, recent developments in DNA testing may make these distinctions irrelevant. With proper testing, most cases are decided on scientific evidence that is almost 100% accurate. Science has effectively removed paternity cases from usual rules of litigation and proof. Now courts and legislatures talk about 97% or 99% certainties instead of terms like "preponderance of the evidence."