As we get deep into the political season, we're all going to be frequently reminded how it is possible to make numbers say just about anything we want them to. Quite frankly, it is not just the arena of politics this happens in. It can be done with all types of math - casino math, included.
By now, many of you well know that a full-pay jacks or better machine pays about 99.5%, which is a very solid number for a casino game. Many of you may even be aware that the Royal Flush contributes 2% of this amount. But what does this really mean? It means that if the machine was defective and NEVER dealt a Royal Flush, but dealt all the rest of the hands in the frequencies we would expect, the payback of the game would be closer to 97.5%. This is about the same payback we would get from a short-pay (8/5) jacks or better machine so should we expect roughly the same experience?
ABSOLUTELY NOT! One of the measures I like to use is what I call a 'session simulator'. This process simulates a session of play for a particular game. For video poker, I use 3 hours of play at 700 hands per hour. For this particular demonstration, I ran 1000 of these sessions under 2 conditions. The first was a full-pay jacks or better machined that NEVER paid a Royal Flush. To be clear, the only way this could ever really happen would be if the machine was broken or rigged. As I don't believe the latter happens in any reputable casino, nor would a broken machine likely stay on the floor for this many hands - this is merely for illustration purposes and to prove a point.
In this scenario, the Player still managed to walk away a winner about 28% of the sessions. This compares to about 29% when a regular full-pay jacks or better is played. Why is there such little impact to this? Under normal circumstances, the Royal would hit only about every 20 cycles or so. Some of these cycles would already be winners, so the Royal Flush doesn't change this. It only changes the magnitude of the win. In the cases where the session was about to be a loser, the Royal most likely flipped ONLY these into winners. However, when we look at the long run, the overall payback of ALL the sessions put together was where we expected it to be - at about 97.5%
When we put the 8-5 jacks or better machine (with the Royal occurring as it should), we find that the Player wins only 14% of his sessions. His winning sessions are cut by half! The overall payback of all the sessions is also what we would expect it to be at 97.5%.
So, why do two different machines paying about the same amount create such different short-term results? This goes to a concept of volatility. There is a mathematical formula for volatility, but I'm afraid if I start explaining it at that level, you're all going to turn the page. That is why I like to use the session simulator as a means of explaining what volatility does and is. When a large amount of the payback is concentrated into a very infrequently occurring hand, there is a larger degree of volatility. In the case of the full-pay jacks or better game without the Royals, I removed a large degree of the volatility. This is why a game with a considerably lower payback that the original version can still have a not very different short-term result.
So, what does this all mean for you? There are two points I'd like you take away from this week's column. The first is to realize how important the Royal Flush is to your long-term results in video poker. If you are on a cold streak of Royals, your short-term results may not look all that different from 'normal', but you may find that your larger bankroll is suffering. If you play for 3 hours at a time, you may find that you're still leaving the casino a winner 3 out of 10 times, but for some reason your wallet still seems a lot lighter than it should. The good news is that in the long run, those Royals will show up as often as they should (assuming you are playing Expert Strategy). Ironically, when the Royals are running hot, you'll still walk away a winner about 3 out of 10 sessions. But, a few more of those sessions will be big winners.
The second point I want everyone to think about is if a 'mere' 800 unit payout occurring roughly every 40,000 hands can make this type of impact to a game, imagine what happens on a slot machine that can pay hundreds of thousands or millions of dollars for a 'hand' even more infrequent. The average payback on a slot machine is ONLY 92-93%. If we consider that many of them will have a massive top pay that might occur only every few hundred thousand hands (or million hands), what % of the overall payback does this account for?
With these occurrences being so infrequent (and COMPLETELY unknown as to how frequent), the payback of the machine without the jackpot could easily be 80-90%. I'd put this through my session simulator but as it is not possible to know the frequency of all the payouts, there is no way to do it. Just for fun, I built an 82.5% video poker paytable and put it through the process and it showed that the Player will walk away a winner only 5% of the time. As we've already shown, it would then be possible to create an infrequent, very high paying jackpot which will push the overall payback up, while barely changing the short-term results.
The end result is one that we know all too well for slots. Very few people walk away a winner even in the short run, which pays for the handful of people who win the big jackpots. I'll take video poker any day!