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Sunday, January 29, 2023

# Special Promotional Payoffs

Sometimes casinos will offer special promotions where they give extra payoffs on video poker hands. These can be quite imaginative:

A box of chocolate-covered Macadamia nuts with each natural four-of-a-kind.

A “card of the day”, where you get an extra \$25 for a natural four-of-a-kind in that card.

50,000 coins (instead of 4,000) if your royal flush is in sequential order left-to-right or right-to-left.

Double payoffs on bonus machines for four aces, between 12:00 and 2:00 am.

Are these worth anything? Remember, you are asking that of someone who advocates joining a slot club for the extra 0.2% return it offers. Of course they are worth something; look for them. Are they worth enough so you can quit your day job? No.

Actually, if you are mathematically inclined, it is usually fairly easy to estimate the minimum value of promotions like these. For example, take the sequential royal flush promotion. The number of different ways you can order five things is 5*4*3*3*1 or 120 ways, so there are 120 different royal flush orders. Two of these are sequential. Thus, 1 out of 60 royal flushes will gain you the 50,000 coins. Thus you could estimate that the royal flush payout is actually increased to (50000 + 4000*59) / 60 or 4767 coins. For any game, you can use my tables to look up how much of your total return is from the royal flush payoff, and increase that by 4767/4000. You will get a figure 0.3-0.4%.

But this is a minimum, because it assumes you do not adjust your playing strategy for the promotional payoff. Playing slightly differently will nudge up the return a bit. So how should you adapt your play? You can pretty much be guaranteed that radical departures from your normal play is not a good idea. However, you can sometimes “hand Slot Gacor calculate” some minor modifications. Again using the sequential royal flush example, you can observe that if you have four royal flush cards in the right place, your chance of drawing the fifth is 1 out of 47. Thus this hand is worth 50000/47=1067 coins, not even counting all the chances at flushes, straights, etc. that you have. Thus, if you had a wild royal flush with deuces, worth 125 coins, discarding the single deuce should be a no-brainer if you could make a sequential royal flush.

If you have three of the royal flush cards in the right place, your chance of filling them in is 1 out of 47*46=2162. (You also have a 1 in 2162 chance of getting them in the other order, for a normal royal flush.) You conclude that the promotional payoff is worth 50000/2162=23 coins for this hand. Thus you would break a pat straight in jacks-or-better (20 coins) to draw for the sequential royal. It turns out you break up a flush (30 coins), too, but that is a close call.

But enough calculations. The point is that these casino promotions are are like slot clubs: they are worth looking for for the additional (albeit small) extra return they give. This is true even if you do not bother to change the way you play. However, if you are able to figure out some small adjustments to your play on your own; so much the better.

The Strategy Sheets

It is possible to memorize a strategy for a particular variation and drill yourself with my computer program until you are confident you are playing accurately. However, I have never had a casino complain when I have referred to a piece of paper to figure out my discard. (On the other hand, bringing a “device for the purpose of cheating” into a casino is a felony in Nevada, so I would definitely not want to test the legality of using a laptop computer.) Be aware that gambling is not an inalienable right in Nevada and casino management can ban you for whatever reason they choose.

Over the years I have tried many formats for the strategy sheets and finally settled on a format that lists the hands in the order in which you should save them, together with visual examples to illustrate the hand and the discard. The examples are picked to illustrate the particularly problematic hands.

To save space, the hand names are abbreviated. These abbreviations should be obvious: RF for royal flush, SF for straight flush, 4K for four of a kind, FH for full house, FL for flush, ST for straight, 3K for three of a kind, 2P for two pair, 1P for one pair, and HC for a hand with only high cards. Incomplete straights and flushes are indicated by number: for example, 4FL is a four-card flush, 3SF is a three-card straight flush.

My abbreviation for a wild card is “w”. Thus, a 1w3K is three of a kind that includes one wild card; in other words, it is a pair plus a wild card. A 2wST is a pat (five card) straight with two wild cards (which, of course, you will not find in the strategy sheets because you should never save it).

The strategies contain some simplifications, but on the whole they are accurate to within 1/100th of a per cent. If you use the strategies when playing my program, the program will occasionally correct your choice. You will notice that this is always happens on hands that are very close choices.

Conclusion

So, where does this all leave you? Can you learn the optimal strategy, find a machine in your favor, and then never work again, just living off your winnings? Suppose you do learn to play optimally, and very fast. You can probably get to the point where you play 10 hands a minute, 600 hands an hour. On a quarter machine, that means you are wagering \$800/hour. If the machine is 0.7% in your favor, that means, on average, you will win \$5.60 for each hour of play. Of course, you are going to slow down occasionally and occasionally make mistakes, so a good estimate is that you could make minimum wage.

Again, this is on average. So how long would it take so you could reasonably expect the good luck and the bad luck to “average out”. 10 hours of play? 20 hours? 100 hours? The true number is much larger. You should play enough hands so that you have a good chance that the frequency of royal flushes you have hit is close to the expected frequency. You should expect to play enough to hit on the order of 100 royal flushes, about 4,000,000 hands, about 4667 high-speed hours, about 582 full 8-hour days–a good two years of full-time gambling with occasional week-ends off.

So, if you think of playing video poker as a career I think you are missing the point. Video poker is recreation, and by learning to play correctly you can make it very inexpensive recreation. Or to be precise, you can make it so that the probability is that it will be inexpensive over the long haul.

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