If there is anything I've learned in about 12 years of doing gaming math professionally, along with another 20+ years of doing it 'unprofessionally', is that the math is completely misunderstood by most. This goes for virtually every aspect of it.
The most common math term for a game is the payback. This is the percent of the total wager that a Player can expect to have returned to him over time. Unfortunately, many people think this means the percent of their BANKROLL (or buy-in) that will be returned. So, if you pull out a $100 and put it into a video poker machine and play a 99.5% game, then you will get back $99.50. I can't say "NO" to this loud enough. It is the percent of the TOTAL AMOUNT WAGERED. So, if you play a quarter machine at max-coin, you are wagering $1.25 per hand. If you play 2000 hands (3-4 hours of moderate pace), then you would have wagered $2500. 99.5% of this is $2487.50. Your loss is $12.50. You'll cash out for $87.50. This does NOT mean the game paid out at 87.5% and you had a bad session.
You may have only put $20 in, and assuming at no point you were down more than $20, then you would get back $7.50. The payback is not determined by how much you buy-in for or how much money you are willing to play. It is determined by the total amount wagered, which is the amount wagered per hand/deal, etc... all summed up.
The other concept that seems to throw a lot of people is when I refer to the 'long run'. When I say a video poker machine pays back 99% and thus in the long run you will lose 1% of your total wager, there are many who have convinced themselves or others that this long run is an entire lifetime and thus you should work on a strategy that has a shorter 'long run'. These people are kidding themselves. Player expert strategy will provide the highest possible payback over the long run. Playing any other strategy will reduce this payback over the long run. Over the short run, anything is possible. You might throw away a Four of a Kind, hold onto a single King and draw the other 4 cards to a Royal. If this actually happens it doesn't mean that you did the right thing, even if it worked out.
How long is the long run? There is no easy answer for that. But the laws or probability say that the longer you play, the closer you get to it. This means the more that you play the likelihood of the actual payback approaching the theoretical payback goes up. If I were to simulate expert strategy play of video poker and take a snapshot of 10 hands at a time, I'll find paybacks all over the place. The 10-hand session that hits a Royal will have a payback of around 8,000%! The 10-hand session that hits nothing will have a payback of 0%. If I do the same but use 100-hand snapshots, I'll find a tighter range of paybacks. The likelihood of a 0% payback is essentially 0. If I go to 1000-hand sessions, the payback range will tighten even more . At 10,000 hands, I will see a range of probably no more than 5-10%, with the theoretical payback roughly in the middle. If I go to 100,000 hands, I'll probably see a 2-4% range. Now, 100,000 hands is a lot of hands, but for someone playing 500 hands per hour, it is 200 hours of play. For someone who goes to Las Vegas a couple of times a year, or who visits a more local casino several times a year, this is perhaps only 2-3 years of play. Not a small amount, but certainly not a lifetime.
Now, if this Player chooses to follow what he thinks is some better 'short' term strategy, he'll find that over this same longer range of time that he will still have a 2-4% range of payback (if I were to simulate this altered strategy). But the center point will be at the lower theoretical payback. So, for example, if someone were to alter his strategy to target Royals, he will increase the probability of hitting a Royal in the next 20,000 hands, but he will lower his overall payback to do so. On any given night, his likelihood of hitting the Royal will go up, but his bankroll will still be decreasing overall at a more rapid rate. Barring some special promotion that pays extra for hitting a Royal, the Player will really just be playing more for pride than for money.
I could list out countless other math fallacies that I have heard through the years. I think the two I have listed today are amongst the tops. The first one is a complete misunderstanding of the terminology and leads Players to abandon strategy thinking it is not working. The second is about Players convincing themselves that some alternate strategy is better in the short run, when the short run is nothing more than a microcosm of the long run. That which works in the long run, will almost always be the right strategy for the short run too. The commonality is that both situations fool the Player into abandoning using the best strategy. This puts a smile on the casinos face!