How can you know an investigation will be fruitless until it is conducted? If Alice trusts Bob, but Eve doesn't, what do we do? Would the investigation into Bob

How can you know an investigation will be fruitless until it is conducted? If Alice trusts Bob, but Eve doesn't, what do we do? Would the investigation into Bob be fruitless or not? Isn't that what the Federal Bureau of Investigation is for? To investigate all claims of impropriety equally and fairly? We accept the false negatives because the cost of giving the benefit of the doubt blindly is a dangerous proposition.