Your committing something very like the Defense Attorney's Fallacy. It's a common fallacy in interpreting statistical likelihoods that involves making a probability judgement as it guilt without taking associative evidence into account.
Defense Attorney's Fallacy
Perhaps the most surprising finding of this research was how easily people can be persuaded to give no weight to associative evidence. The associative evidence presented in the two experiments was quite powerful: a match between the suspect and perpetrator on a characteristic found in only 2% (Experiment 1) or 1% (Experiment 2) of the population. According to Bayes' theorem, a person who initially thought the suspect's probability of guilt was .10 should revise that estimate upward to .85 and .93, respectively, in light of this associative evidence. Of course, one need not rely on Bayes' theorem to conclude that the Defense Attorney's Fallacy is inappropriate. It is difficult to imagine any normative model of judgment that would give no weight to associative evidence.Yet many people in these experiments were persuaded that because a large number of individuals other than the suspect would also "match" on the relevant characteristic, the "match" is uninformative with regard to the suspect's likelihood of guilt. As noted earlier, what this reasoning ignores is that the overwhelming majority of the other people who possess the relevant characteristic are not suspects in the case at hand. The associative evidence drastically narrows the class of people who could have committed the crime, but fails to eliminate the very individual on whom suspicion has already focused.
The argument favoring the Defense Attorney's Fallacy was particularly persuasive. In Experiment 2, over 60% of people who heard the argument were persuaded that the associative evidence deserved no weight. Finding some way to combat this powerful fallacy is clearly an imperative for future research.
Source: Interpretation of Statistical Evidence in Criminal Trials: The Prosecutor's Fallacy and the Defense Attorney's Fallacy Thompson, William C. ; Schumann, Edward L. Law and Human Behavior, 1 September 1987, Vol.11(3), pp.167-187