Here is what I think is a better example of the Gettier problem, and a subsequent reason the Gettier problem is flawed in its definitions of truths.
You are driving down the highway, passing what appear to be several dozen barns. Unknown to you, all but one of these barns is a stage prop cutout. You decide to stop at one of these barns and by luck it is the only real one. You now have a belief (which is that the barns you see are real), which is justified, and in this case, true. But it cannot be called knowledge. Why? Because the belief is imprecise and leaves room for vagaries. A belief should describe the fundamental mechanisms of the universe. i.e. the presence of light patterns in format X indicates structure Y, because light interacts in ways Z. In this case the belief about the barns is unjustified and untrue, because there is an additional way format X could be created, by structure Y2 and light interaction Z2 (the cutout). Discovery of the real barn is only weak evidence for the belief that format X indicates a real barn, as the discovery proves the possibility thereof, but does not eliminate the alternative (cutouts). Under this new definition of belief, a concept of the universe fundamental mechanisms, as opposed to informal correlations, only accurate and precise beliefs that allow prediction generation constitute knowledge.
But this is not how we think. And for very good reason. Typically a scenario in which all options appear identical to cursory examination, and in which detailed examination provides some conclusion about one option, it can be a huge waste of time and effort to generate all theoretically possible contradictory scenarios and test them, not to mention the possibility that you may not think of or be able to test all such options. So our brain takes a mental shortcut. Barns appear to be same? Check. Barn 10 is a 3d barn? Check. Therefore all barns are 3d. Though nothing was falsified, it is a useful informal deduction which only fails us in extreme circumstances such as the problem listed above. But there is a very good reason that we don't use such logic in scientific experimentation. When we have not repeatedly experienced a phenomenon and have no hard-set reason to believe a correlation indicates causation, falsification is all we can trust. We don't have the huge backdrop of everyday data to fall back upon. Oftentimes we have a hard time realizing this though, and make assumptions as if we have such a backdrop when we don't.
Scientific Method: Don't do that.
One of the few things that I really appreciate having encountered during my study of philosophy is the Gettier problem. Paper after paper has been published on this subject, starting with Gettier's original "Is Justified True Belief Knowledge?" In brief, Gettier argues that knowledge cannot be defined as "justified true belief" because there are cases when people have a justified true belief, but their belief is justified for the wrong reasons.
For instance, Gettier cites the example of two men, Smith and Jones, who are applying for a job. Smith believes that Jones will get the job, because the president of the company told him that Jones would be hired. He also believes that Jones has ten coins in his pocket, because he counted the coins in Jones's pocket ten minutes ago (Gettier does not explain this behavior). Thus, he forms the belief "the person who will get the job has ten coins in his pocket."
Unbeknownst to Smith, though, he himself will get the job, and further he himself has ten coins in his pocket that he was not aware of-- perhaps he put someone else's jacket on by mistake. As a result, Smith's belief that "the person who will get the job has ten coins in his pocket" was correct, but only by luck.
While I don't find the primary purpose of Gettier's argument particularly interesting or meaningful (much less the debate it spawned), I do think Gettier's paper does a very good job of illustrating the situation that I refer to as "being right for the wrong reasons." This situation has important implications for prediction-making and hence for the art of rationality as a whole.
Simply put, a prediction that is right for the wrong reasons isn't actually right from an epistemic perspective.
If I predict, for instance, that I will win a 15-touch fencing bout, implicitly believing this will occur when I strike my opponent 15 times before he strikes me 15 times, and I in fact lose fourteen touches in a row, only to win by forfeit when my opponent intentionally strikes me many times in the final touch and is disqualified for brutality, my prediction cannot be said to have been accurate.
Where this gets more complicated is with predictions that are right for the wrong reasons, but the right reasons still apply. Imagine the previous example of a fencing bout, except this time I score 14 touches in a row and then win by forfeit when my opponent flings his mask across the hall in frustration and is disqualified for an offense against sportsmanship. Technically, my prediction is again right for the wrong reasons-- my victory was not thanks to scoring 15 touches, but thanks to my opponent's poor sportsmanship and subsequent disqualification. However, I likely would have scored 15 touches given the opportunity.
In cases like this, it may seem appealing to credit my prediction as successful, as it would be successful under normal conditions. However, I think we perhaps have to resist this impulse and instead simply work on making more precise predictions. If we start crediting predictions that are right for the wrong reasons, even if it seems like the "spirit" of the prediction is right, this seems to open the door for relying on intuition and falling into the traps that contaminate much of modern philosophy.
What we really need to do in such cases seems to be to break down our claims into more specific predictions, splitting them into multiple sub-predictions if necessary. My prediction about the outcome of the fencing bout could better be expressed as multiple predictions, for instance "I will score more points than my opponent" and "I will win the bout." Some may notice that this is similar to the implicit justification being made in the original prediction. This is fitting-- drawing out such implicit details is key to making accurate predictions. In fact, this example itself was improved by tabooing[1] "better" in the vague initial sentence "I will fence better than my opponent."
In order to make better predictions, we must cast out those predictions that are right for the wrong reasons. While it may be tempting to award such efforts partial credit, this flies against the spirit of the truth. The true skill of cartography requires forming both accurate and reproducible maps; lucking into accuracy may be nice, but it speaks ill of the reproducibility of your methods.
[1] I greatly suggest that you make tabooing a five-second skill, and better still recognizing when you need to apply it to your own processes. It pays great dividends in terms of precise thought.