Can you have knowledge that is based on a false belief?

Suppose you have two kids: A and B, and you believe that A is left-handed, and that B is left-handed. Being a deft reasoner, you conclude: At least one of my kids is left-handed. Now suppose that, really, B is not left-handed (though you're right, and knowledgeably so, about A). It seems to me that it is still right to count you as knowing that at least one of your kids is left-handed, and it also seems plausible that your belief is in some good sense "based on" a false belief (that B is left-handed). Someone might challenge the latter claim, by observing that your belief about B is inessential to your justification for the "at least one" belief---if you hadn't believed that B was left-handed, after all, you still would have been justified in concluding that at least one of your kids was left-handed. But on that way of viewing things, your "at least one" belief is based on neither of your two beliefs about the left-handedness of A and B (for each of them individually is such...