Likelihood ratios measure the strength of forensic deoxyribonucleic acid (DNA) evidence supporting the hypothesis that a given suspect is the source of a matching sample, versus the alternative that someone else is the source. One component of the DNA evidence consists of samples from the scene of the crime and from the suspect. Another consists of samples from a reference population. Treating each of these components as being generated by random (probabilistic) processes, and measuring the evidence by likelihood ratios, we examine the probability of obtaining strong implicating evidence when the given suspect is indeed the source of the crime scene sample and when someone else is the source. (In the latter case, the evidence is misleading.) The distinction between the strength of the evidence and the improbability of its occurrence is critical, as is the importance of conditioning on the particular circumstances of a given case to ensure the relevance of the calculated probability of misleading evidence.