Royal even beats Expected Value theory

One of the reasons that people play slots is the very large paybacks that some of them offer. It is not uncommon for a slot machine to offer 10,000 or 100,000 or sometimes even millions in a single spin of the reels.
The fact that the odds of hitting the exact combination required could be hundreds of millions to one does not factor in much to the decision making process. This has always been the theory behind the Lotteries played in many states. The higher the single win potential, the lower the payback the player is willing to accept.
As long as the outcome has the most remote chance to forever change one’s life, the player throws the concept of Expected Value right out the window.
On a slightly lower level, this can explain why some players will trade in some amount of overall payback for the opportunity to hit a Royal Flush in video poker. The Royal offers a nice payback of 4,000 at max-coin, but this is not exactly life altering. But, it does make a good photo-op and some good story telling.
If you follow expert strategy for a Jacks or Better machine, you’ll be rewarded with a Royal about once in every 40,400 hands. The key phrase is if you follow Expert Strategy.
When my father developed the strategy for Jacks or Better, there were some surprises. One of the first and biggest was that you do not hold 3 high cards over 2 suited. If you have Jack and King of Diamonds and the Queen of Clubs, you discard the Queen of Clubs. This is due to the powerful payout of the Royal.
It is quite a long shot to hit one needing 3 cards, but the over-sized payout helps to make this the right decision. In the meantime, you’ll lower your chances to hit a Straight, but increase the likelihood of many other hands (like a Flush!).
At the other end of the spectrum, it was discovered that the player should never hold a suited 10-Ace in Jacks or Better. The 10 is just too weak of a card relative to one that is a jack or better. The odds of a Royal is the same as any other 2-card one, but it is harder to fill a Straight of 10-A than a 10-J or 10-Q. This is just enough damage to make holding the single Ace the proper play, even though it eliminates completely the chance for a Royal.
If your goal is only to hit a Royal, you could simply discard just about any hand in favor of a partial one. If dealt a suited J-A along with another jack, just discard the off-suit jack and keep the 2-Card Royal.
Play like this and you can make a Royal appear nearly twice as often as it is supposed to, but at great peril to your bankroll. If money is no object and you just want a picture of the Royal you hit, I guess this would be the plan for you.
If, however, you are like the rest of us and might be willing to give up a little payback (say 0.5%) for increasing the frequency of a Royal by around 20-25%, then I have a different suggestion. By making 4 changes to expert strategy to jacks or better, you can achieve what I just described. These changes are as follows:
Never hold more than one low face card if you are dealt more than one face card and they are not suited. So, if dealt an unsuited J-Q-K, just hold one of them (the lowest of them would be ideal as it increase the probability of a Straight).
Play the suited A-10 (obviously, not over Low Pairs, etc…)
If you are dealt a 3-card Inside Straight Flush with a single face card, play only the face card. If no face card, play as a 3-Card Inside Straight Flush.
If dealt a 4-Card Inside Straight with high cards, keep only the suited face cards. Keep in mind, this might mean keeping only one high card.
To be clear, this is not to be considered an alternative to Expert Strategy on any sort of regular basis. You will be doubling the house advantage for the game. But, if you just need to have a Royal, this will increase your chances without destroying your bankroll.
While this will push the Royal Frequency to about 1 in 30,000, keep in mind that it will not be uncommon to still go 40,000 or even 60,000 hands before you hit one.
Also, the above strategy assumes that you otherwise are playing Expert Strategy properly, in order to achieve the results described.

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